Virginia Lt. Gov. Richard J. Davis, responding to a groundswell of support from the Democratic Party's rank and file, said late tonight he would enthusiastically accept a U.S. Senate nomination that appeared his for the asking.

After an hour-long, closed-door meeting among party leaders, Davis broke a two-week silence that had paralyzed the party on the eve of its nominating convention. Although the overwhelming favorite of party regulars, the affable white-haired mortgage banker from Portsmouth only two weeks ago had taken himself out of the nomination for the U.S. Senate seat now held by Sen. Harry F. Byrd Jr.

As the party was unable to settle on another nominee of statewide prominence, sentiment for Davis built up dramatically. A well-orchestrated draft-Davis boom swept the convention when it began this evening, as supporters passed brown paper bags to collect the $1,223 needed to place their candidate's name in nomination. Meanwhile, backers of other declared candidates--including former congressman Joseph Fisher and Fairfax County prosecutor Robert Horan--were visibly angered at Davis' continued indecision.

But as the pressure built, Davis finally agreed to declare his intentions.

"It is only fair that I make known to the other candidates and to the delegates my intentions to accept the nomination if chosen," he said in a written statement. "In light of this situation, I am a candidate and would welcome enthusiastically the nomination of my party."

"State Del. Alson Smith of Winchester, a top party fund-raiser, remarked that the stage seemed set for "a true draft."

Although Horan, Fisher and other candidates were included in the meeting of party leaders, no one made any move tonight to back out of the race. "I'll have to reflect on that," Fisher said. "As of now my plans have not changed."

Former Attorney General Andrew Miller became the latest candidate to surface at the convention when his supporters began circulating petitions tonight to place his name in nomination. Miller declined comment on Davis' statement.

The strategy of the Davis forces was a tricky one. When Davis bowed out of the race two weeks ago there were complaints that it would appear unseemly for him to seek another office less than a year after campaigning for the lieutenant governor's slot. In addition, Davis had gotten a lukewarm reception from the state's conservative fund-raisers, many of whom viewed him as too liberal for Virginia.

Hence Davis found himself in a box. By removing himself from the race, he had encouraged several other Democrats to jump into the race. All had invested money and time in last-minute, hastily thrown together campaigns. As Davis continued his silence, frustrated backers of the announced candidates were seething.

In mid-afternoon, one furious Horan coordinator erupted in anger at Cunningham in a locker room behind the convention podium, shouting that his camp was "fed up with Davis . . . he doesn't even have the courtesy to call us over the phone."

Among the most organized was the 68-year-old Fisher, who appeared before a labor caucus to stress his six years in Congress. "It's not academic with me," said Fisher. "It's not how I would vote--it's how I have voted."

Among the other candidates in race were state Sen. Virgil Goode Jr., a 35-year-old self-styled populist whose hand-lettered signs "Feeling Good for Goode" gave a taste of his homespun campaign style. Less well known but equally tenacious was Peter J. Tsakanikas, an Arlington entrepreneur, who used his campaign pamphlets to tout his latest business venture -- a talking computer. "Sure, his product gets exposure, but his product is outstanding," said Jim Massey, campaign manager and one of four known Tsakanikas delegates, who include Tsakanikas and his wife.

The convention formally kicked off tonight with a speech by Gov. Charles S. Robb, who alluded to the Democrats' two-month ordeal in attempting to find a Senate candidate acceptable to all factions in the party.

One by one, a series of ranking Democrats took themselves out of the race, starting with the withdrawal early last month of Del. Owen B. Pickett of Virginia Beach, the man picked by Robb and other leaders as the party's concensus candidate. In the chaos that ensued, Robb was criticized for failing to take charge of the party and heading off the kind of internal bickering that has plagued Democrats in Virginia.

"It has been admittedly a difficult and trying period for all of us," said Robb tonight, adding that the party's travails had been "character building."

The governor's remarks were part of his introduction of retired Lt. Gen. Samuel V. Wilson, a decorated war hero and intelligence expert whom Robb had chosen as the convention's keynote speaker. Wilson attacked the Reagan administration's economic policy. "Republicans say 'A rising tide lifts all yachts,' " he said to laughter and applause.