WASHINGTON -- John Kerry's choice of John Edwards as his running mate was received favorably by the public, polls suggest, but it has made little difference so far in the race with President Bush.

Kerry strategists are trying to lower expectations for a "bounce" in the polls that presidential candidates sometimes get after choosing a running mate or attending a convention.

Bush strategists were quick to raise expectations of a double-digit "bounce" for the Kerry-Edwards team by the end of the Democratic National Convention.

Tad Devine, a Kerry campaign strategist, said he does not believe Republican claims about "a double-digit" bounce of 12 percentage points to 15 percentage points.

"We've gotten the bounce already that we're going to get," Devine said. "If you look at the Democratic vote, it has already consolidated behind John Kerry."

Both Devine and Bush campaign strategist Matthew Dowd said on "Fox News Sunday" that they have detected slight gains for Kerry in the polls after the Edwards choice.

Kerry's announcement Tuesday was followed by a tour of several states by the candidates and their families.

Kerry has "gotten a slight uptick, whether it's temporary or not," Dowd said.

But those looking for a Kerry surge in the polls after the Edwards pick saw a shift of a few points, often within a poll's margin of error.

An AP-Ipsos poll released Thursday offered an early hint there would not be a post-Edwards bounce for Kerry.

Bush had a slight lead over Kerry as voters expressed increasing confidence about the economy. Bush was at 49 percent, Kerry at 45 percent and independent Ralph Nader at 3 percent, according to the poll conducted for the AP by Ipsos-Public Affairs.

Other polls in the next few days showed Bush and Kerry essentially tied in a three-way contest. Kerry had a slight edge in a two-way race with Bush in some polls.

Even though the race remains close, weekend polls found encouraging news for Edwards.

Almost half, 47 percent, said in a Time-CNN poll that Edwards would make a better president than the current vice president, Dick Cheney, while 38 percent said Cheney would be better.

When people were asked in a Newsweek poll who they would pick if they could vote separately for vice president, they chose Edwards by 52 percent to 41 percent for Cheney.

Asked last week about Edwards' campaigning and personal skills, the president was asked at a news conference last week how Edwards compared with Cheney. "Dick Cheney can be president," Bush said quickly.

In an interview done for "60 Minutes," Kerry said of Edwards: "He is more qualified, more prepared in national affairs and national issues than George Bush was when he became president."