Attorney General William French Smith and about two dozen drug enforcement and immigration officials will leave Tuesday on a "fact-finding" mission through Asia and Europe to discuss international drug trafficking, refugees and international terrorism problems.

The stops on the trip will include Tokyo, Hong Kong, Thailand, Pakistan, Paris and Rome, according to a senior administration official, who asked not to be identified.

About two-dozen people will go on the trip, including Drug Enforcement Administration Administrator Francis M. Mullen Jr. and Immigration and Naturalization Service Commissioner Alan C. Nelson.

An official said the trip was intended to increase cooperation between governments in fighting international drug smuggling and in handling refugee resettlement. The official said there is no figure yet for what the 22-day trip will cost.

In Tokyo, Smith will talk with Japanese officials about the seriousness of the recent indictments of officials from Hitachi and Mitsubishi for conspiring to steal computer secrets from IBM. The indictments have been portrayed in the Japanese press as more political than criminal.

Smith plans to spend a day in Paris talking to officials about proposals for dealing with international terrorism.

In Rome, where he will spend four days, he will meet with government officials to talk about heroin processing and will visit the Vatican. Officials said 90 percent of the heroin coming into the United States is processed in Italy.

Visits to the other countries will focus on refugees and the production of opium poppies. Smith will talk about programs to substitute other crops for the poppies.

Officials said the Smith trip will be the first international trip by an American attorney general to discuss narcotics and refugee problems.

He will be accompanied by his wife, Jean. An official said the government will pay for her trip since she plans to do some work for the U.S. Information Agency. An Air Force plane will be used for the trip.

The trip comes at a time when the administration is attempting to place greater emphasis on investigating organized crime and illegal drugs. On Thursday, President Reagan announced a new program to set up 12 regional task forces and spend up to $200 million in additional funds to investigate illegal drug trafficking and organized crime.