Richard E. Lyng, whose loyalties to President Reagan are rooted in a political relationship of more than 20 years, offered to resign as agriculture secretary last spring in an effort to win a key Republican senator's support of the president's veto of a controversial highway bill.

Lyng's move, obviously, did not work. Sen. Mitch McConnell (R-Ky.), who was angry at Lyng for making changes in the burley tobacco support program, ignored the offer and cast a crucial 67th vote that allowed the Senate to override Reagan's veto on April 2.

McConnell's resentment toward the secretary and his department still runs deep. McConnell is blocking Senate confirmation of Ewen Wilson as assistant secretary for economics and Milton Hertz as administrator of the Agricultural Stabilization and Conservation Service, which oversees the tobacco program.

McConnell, who voted against the nominations at the Agriculture Committee, has given no indication if or when he will remove his objections. He has told reporters only that he has concerns with USDA's "insensitivity" toward his state.

Neither Lyng nor McConnell could be reached for comment on this matter, but associates of both men confirmed that the secretary made his unusual offer to resign in a tense meeting in the office of Senate Minority Leader Robert J. Dole (R-Kan.) not long before the override vote.

"It was very tense . . . the Hill was overwhelmed with the pressure on the veto override," one Republican source said. "That the secretary was willing to sacrifice himself showed how loyal he is and how committed the administration was to winning that vote."

Lyng's offer followed a meeting with Reagan, Dole and McConnell in the Oval Office, where Reagan tried to win the Kentucky senator's support for the veto. Despite Reagan's opposition, McConnell in February was among 96 senators who voted for the highway bill.

Sources said McConnell told Reagan he would not change his vote "because of this man," pointing at Lyng. McConnell was particularly angry about a department decision last year to lower the price-support loan rates for certain types of burley tobacco, a Kentucky specialty.

"It was a dumb thing for the administration to do politically," a GOP source said. "McConnell had lobbied hard against the change and he felt this administration could have helped him. In the end, the loan rate change had no effect at all, except to hurt Mitch's credibility at home."