President Reagan and House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) played a wary game of chicken yesterday as each awaited a move by the other on a politically sensitive Senate proposal for tax increases and Social Security cutbacks to help reduce budget deficits.

But the White House avoided criticism of the Senate proposal, and O'Neill, after an early-morning broadside against the plan, later issued a conciliatory statement that he would "go the extra mile" to get an agreement.

"If President Reagan agrees to all or even part of the budget proposal made by the Senate Republicans yesterday, then the House will sit down and negotiate on the basis of that budget," O'Neill said after consulting with lawmakers who were anxious to get a budget agreement and fearful that Democrats would be blamed if the effort failed.

O'Neill's statement appeared as the one bright spot in an otherwise rancorous, tension-filled day fraught with gloomy talk of a stubborn confrontation between Reagan and O'Neill that could lead to collapse of the deficit-reduction effort.

It's "high noon at the old corral," said Senate Majority Leader Robert J. Dole (R-Kan.) before O'Neill issued his conciliatory statement.

"We're out in Dodge City and we've got Tip O'Neill on one end of the street, Ronald Reagan on the other. They're looking at each other trying to decide who is going to blink first."

"They're both going to have to blink together," said Senate Budget Committee Chairman Pete V. Domenici (R-N.M.), whose proposal for summit-style budget negotiations at the White House was left dangling in the wake of the dispute over who should make the first move. Although O'Neill's later comments were taken as an encouraging sign of a possible breakthrough, it was not clear last night what the next step might be.

There was no immediate response from the White House, and no further negotiations have been scheduled.

At stake were $338 billion in deficit reductions over three years that were proposed Thursday by a bipartisan team of Senate budget negotiators, including a $5-a-barrel oil-import fee and a delay in inflation adjustments for both income taxes and Social Security benefits.

House negotiators took the proposal under consideration but expressed qualms about serious negotiations on either taxes or Social Security so long as Reagan has removed both issues from the bargaining table.

It was Reagan's agreement with the House to abandon a Senate proposal to freeze Social Security benefits for a year that led to a virtual collapse of the talks last week, angering senators and prompting them to put both Reagan and O'Neill on the spot by including both tax increases and Social Security trims in their new offer.

Reagan is adamantly opposed to tax increases, and O'Neill is equally opposed to tampering with Social Security, but the two adversaries appear reluctant to challenge one another on either issue.

Frustrated with both of them and smarting over suggestions that they were to blame for breaking up the negotiations, the senators came up with the biggest deficit-reduction package yet to go on the table and shifted the burden of decision back to Reagan and O'Neill.

Dole opened fire yesterday morning by suggesting that O'Neill and House Majority Leader James C. Wright Jr. (D-Tex.) "stop dumping on the president and start looking at their own responsibilities."

O'Neill, said Dole, tried to shoot down the Senate proposal before it was even announced.

"I don't see any reason for the president to rush to have another meeting. I'm waiting for leaders on the House side to indicate they're willing to negotiate."

As for Reagan's opposition to the oil-import fee on grounds that it is a tax, Dole said: "It is a fee, an F-double-E. It's necessary for national security, conservation, preservation of industry and other things -- to save the country."

O'Neill accused Senate Republicans of "stubbornness" and "ridiculous" action in proposing a plan that the president didn't support. "I'm beginning to question whether they really want a budget at all . . . ," he said.

To Reagan, he said, "Now it is time for the leader of the Republican Party to tell his teammates to drop the gimmicks and reach a final budget agreement." From the White House came the word that the next move was up to the House. "It is now time for the House to consider and respond to it," presidential spokesman Larry Speakes said.

Asked if that meant that the president will not say anything until the House does, Speakes said, "We would like to see some response from the House and at the same time indicate that, when there is some sort of agreement on Capitol Hill, we would be prepared to sit down with the bipartisan leadership and discuss proposals . . . . We're ready once there is some sort of jelling on the Hill."

O'Neill, in his second statement, continued to insist that Reagan go first but hinted that the House would be willing to negotiate based on even a suggestion of support from the president.

"If the president is willing to broach the subject of taxes, we'd be willing to talk about the form of taxes," said an aide. While still opposed to tampering with Social Security inflation adjustments, O'Neill remains interested in his earlier proposal, dumped by House budget negotiators, to tax the benefits of wealthy Social Security recipients, the aide said.