The U.S. Chamber of Commerce kicked off the second round of its campaign against administration-backed legislation to overhaul labor laws yesterday with a charge that the bill's passage would "strong-arm" 60 per cent of the nation's labor force into unions.

Chamber officals predicted at a press conference that the legislation would triple the current level of unionization - now 20 per cent and falling - during the next decade.

The AFL-CIO, which has made no such claims and contends the bill simply puts teeth in the 42-year-old National Labor Relations Act, countercharged that the Chamber's claim is a measure of the success of efforts by employers to thwart the original intent of the law.

"It's an excellent argument for passing the legislation, said AFL-CIO spokesman Albert J. Zack.

The Chamber and other business groups suffered a major setback when the House approved the bill last October with only minor modifications. A tough battle - and probably a filibuster - faces the legislation when it comes to the Senate floor, possibly as early as next month.

The bill which the Carter administration's main gift to organized labor last year, generally would make it easier to organize unions and win contracts by speeding up representation elections and strengthening penalties for unfair labor practices.

The Chamber's press conference was called to release results of a December poll it commissioned from Opinion Research Corp. of Princeton, N.J., on labor issues.

The results showed most repondents thought unions were too powerful and had too much influence with Congress, but offered somewhat contradictory evidence on labor legislation.

While only 22 per cent favored legislation that "makes it easier than it now is for unions to organize nonunion employees," legislation making it easier to "organize the employees of small business" was approved, 48 to 45 per cent.

The poll showed that most people support the Chamber's position on various aspects of the bill. In a previous AFL-CIO poll, with questions worded differently, most poeple supported organized labor's position.