The White House has enlisted the services of Lloyd N. Cutler, a prominent and influential Washington lawyer, to coordinate the administration's efforts to sell the new strategic arms limitation treaty (SALT II) to the Senate.

Cutler, 61, has built a reputation as the sort of lawyer business executives turn to when they have a serious problem. His firm, Wilmer Cutler & Pickering, is one of Washington's Largest and most prosperous.

Cutler will stay with the firm while serving as a special counsel to the president for SALT. He will not receive any government salary.

Cutler will fill a role that has gone begging in the Carter administration thus far - field commander for the troops that are hoping to persuade 67 senators to endorse the new arms pact. It is understood that White House officials have recently concluded they had to have one senior person who could undertake that job.

Cutler has already performed numerous tasks for the Carter administration, and Defense Secretary Harold Brown recently urged him to accept the No. 3 job in the Pentagon, under-secretary for policy. Cutler declined that offer, and Brown then encouraged the idea of this new role as coordinator of SALT selling.

Cutler is also an old friend of Cyrus R. Vance, the secretary of state.

Reached by telephone yesterday, Cutler confirmed that he had been "asked to come aboard to help organize the presentation" of the administration case for SALT. He and several White House officials all discussed the job as comparable to the position of a senior lawyer in a big case.

The administration will begin formal presentation of its SALT case on July 9, when the Senate Foreign Relations Committee will open hearings on the treaty. Later in July the Armed Services Committee will also hold hearings.

Cutler met yesterday with Vice President Mondale, Robert G. Beckel, the SALT lobbyist on the White House staff, David Aaron of the National Security Council and an interagency committee of officials who have been preparing SALT testimony.

Senators sympathetic to SALT II have complained for months that there was no single, authoritative figure in the administration to whom they could turn to talk about the SALT treaty. Administration officials are said to feel th* at Cutler could assume this role if he develops the right relationships with other important figures in the government.

It could also fall to Cutler to devise negotiating strategies with uncertain senators who seek special considerations in return for their support. CAPTION: Picture, LLOYD CUTLER . . . new liaison with Senate