Republican George Bush is ready to make his first cut in choosing a running mate after making contact with a core list of more than 12 possible contenders.

Those already reached by Bush range from former rivals Sen. Bob Dole of Kansas and Rep. Jack Kemp of New York and Dole's wife, former Transportation Secretary Elizabeth Dole, to several whose names did not figure in earlier speculation including Sen. Dan Quayle, R-Ind.Quayle spokesman Jeff Nesbit said a visit from Washington lawyer Robert M. Kimmitt, who is overseeing the screening process, followed a call from the vice president. "He would accept it if asked," Nesbit said.

Sen. Pete V. Domenici, R-N.M., also confirmed that he had been asked by the vice president to submit background information.

Domenici said he was honored and proud to be considered, but said in a statement, "I have not been seeking this position."

Bush, who has been busy making phone calls this week and has not had many campaign appearances. He planned to visit a corporate child-care center Friday in Tysons Corner, Va., a suburb of Washington.

"He wanted to follow up on his child-care initiative," said Bush spokeswoman Sheila Tate. Bush on Sunday proposed a $1,000 tax credit for child care primarily aimed at low-income working mothers.

The Bush campaign is stressing the issue in part because of concern about the "gender gap"; polls show him trailing far behind Democratic nominee Michael Dukakis among women voters.

Bush has personally gotten in touch with nearly everyone on his "long list" of vice presidential candidates, aides said. "I would suspect that process is complete," said one aide. However, Bush could decide to make more calls, the aide said.

Campaign sources said Bush's list to date also includes Sens. Alan Simpson, R-Wyo.; William L. Armstrong, R-Colo.; John Danforth, R-Mo. and Nancy L. Kassebaum, R-Kan.; Govs. Thomas H. Kean of New Jersey, John Sununu of New Hampshire, Carroll A. Campbell of South Carolina, George Deukmejian of California; and former Govs. Lamar Alexander of Tennessee and Richard Thornburgh of Pennsylvania, who recently was nominated by President Reagan to be attorney general.