As if the Republicans did not have enough trouble in the Midwest, where incumbent governors are bowing out in Iowa, Minnesota, Michigan and Ohio, there are signs of trouble gathering for Illinois Gov. James R. Thompson (R) as well.

Polls show Thompson out front of his challenger, former senator Adlai E. Stevenson III (D), but lately Thompson is enmeshed in controversy. Many eyebrows were raised in normally Republican legal circles when Thompson publicly lobbied Sen. Charles H. Percy (R-Ill.) to add the name of the governor's wife to the list of persons recommended for appointment to a vacant federal district judgeship.

Jayne Carr Thompson, 35, graduated from law school in 1970 and spent seven years in the Illinois state attorney general's office, quitting practice in 1977 when her husband became governor and she became pregnant. The Chicago Tribune quoted sitting judges and law school officials as being critical of the governor's pushing her candidacy.

Then last week, Thompson called a press conference to rebut charges of impropriety in use of state planes for two trips from Springfield to a northern Illinois resort, where he and his wife spent weekends as nonpaying guests of the management. Thompson said he would not reimburse the state because he "spent time on state business" during the golfing weekends, which he said helped promote the state's tourist facilities.