Poland's opening to Western market forces has brought an unexpected side effect: a booming traffic in the country's blond, blue-eyed babies.

Since the fall of Communism two years ago, Western embassies in Warsaw have reported a striking rise in the number of residence visas and passports granted to Polish infants and toddlers.

Polish officials say that many of the adoptions are legal but that the black market is growing. And participants in such transactions say some young mothers are being pressed to sign away the rights to their children. In some cases, officials say, poor, pregnant women give up their babies in exchange for money directly. But most often, they say, administrators of homes for single mothers, as well as the attorneys involved in the adoptions, receive up to tens of thousands of dollars.

Reports that large amounts of money have changed hands in exchange for babies are not new in Eastern Europe or the third world: Romania became notorious for the practice after its 1989 revolution.

But the issue is potentially explosive in Poland because the competition from foreigners keeps Poles from adopting Polish children and because some of the cases reported are linked to the Roman Catholic Church. No Accurate Count

Barbara Passini, director of the state-run adoption organization, the Children's Friendship Agency, said: "There many be several hundred, several thousand, maybe even tens of thousands of cases. There is no way to know. I hate to say it, but it seems to me that Poland has one of the most serious markets of white babies. It sickens me to use this term, but unfortunately it is the truth."

Mrs. Passini said there was no way of knowing how many illegal cases there were, and added that throughout the 1980's, adoptions of Polish children by Westerners totaled less than 100 annually.

But last year, for example, the American Embassy granted American citizenship to 96 Polish children, a 40 percent jump from 1990. Swedish officials said there were 112 adoptions last year, while the French Embassy granted 115 passports and the Italians, 70. No one can say how many more children left without proper papers.

Consular officials said they try to make sure that legal requirements are followed, but acknowledge that certain improprieties -- large sums of money in exchange for babies -- are nearly impossible to police.

Pioneering reporting on this subject has been done by Nie, a weekly journal run by Jerzy Urban, the once-despised spokesman of the Communist Government. While several Polish newspapers have written about the issue, most have shied away from the church's role, which Nie has covered. 'Great Moral Discomfort'

In a recent article, Marek Baranski wrote about one young woman in the city of Lublin who gave her unborn child up for adoption to an American couple in December 1991 after being pressed by the nuns caring for her in a church home for single mothers.

Since the article appeared, Mr. Baranski said he has received several dozen letters, most of them anonymous, from women throughout Poland who wrote of having the same treatment in church-run homes.

According to the Nie article, the young woman, who was not identified by name in the article, said that the mother superior of the home received up to $25,000 for each baby boy and $15,000 for each baby girl. The woman, who is now in an alcohol rehabilitation center in northern Poland, is not allowed to have visitors, but she reasserted her account in a telephone interview in March.

Two visitors driving a foreign car went to the home named in the Nie article not long ago. The mother superior at the home, Sister Benigna, greeted the visitors with blessings and proudly displayed her papal award for "defending life," an honor Pope John Paul II bestows on anti-abortion crusaders in his native Poland.

"How can I help you, dears?" she said, offering tea. When the two said that they were journalists, Sister Benigna rose to her feet.

"There was a very bad article about us," she said. "It has given us great moral discomfort. I cannot give you any information. Good-bye." Fear and Church's Role

She acknowledged having "helped" several foreign couples in adoptions, but denied that the home had ever received money for its troubles.

A spokesman for the Polish episcopate, the church's headquarters, declined to comment on the church's role in foreign adoptions.

An adoption official in Lublin who insisted on not being identified said of the nun's denial: "Those are lies. People know about it. I wouldn't hesitate to say everyone involved in adoptions in this country knows about it. But no one wants to speak up. You see, for 45 years we were scared of the Communists. Now it is the church that is controlling Poland, and again we are afraid, even though we are Catholics."

A 28-year old Lublin woman, who would give her name only as Iwona, told of her treatment at Sister Benigna's home five years ago, when she was pregnant with her son, whom she is raising. A $30,000 Process

"I was treated very well and with compassion until it became clear I was not going to give up my child," she said. "Then I became a second-class citizen. I had to share a room with three other girls -- all of us were keeping our babies -- and Sister would wake us in the middle of the night to tell us how worried she was about us, how God told her how bad and difficult our lives would be. You could call it brainwashing. It was frightening. To be pregnant and homeless and alone is frightening enough. Then to have this old woman who at first treated you well come in and scare you at night, well, it's unthinkable."

A Westerner who lives in Poland said he helped a Western couple adopt a Polish infant from a church-run home that was in such poor condition that excrement was smeared on the bathroom walls. He agreed to describe the process only if his name and nationality were not revealed.

He said the couple spent more than $30,000 in the seven-month process on flights to Poland, lawyer's fees, and a "donation" to the priest in northern Poland. The man said the "donation" was near $10,000.

"And after that the priest bowed his head, looked dejected, and said, 'Is that all?' " the man said.

Mrs. Passini said she expected the transactions to increase.

"Sadly, this problem will continue," she said. "We live in a country where poverty and misery climb out of every window. What kind of countries give up their babies, their futures, for adoptions? Only poor ones. Until every Polish family is eating bananas every day, this will go on."