Democratic officials said that with Mr. Obama heading into battles over the federal budget, a plunge into the fray over public-sector collective bargaining could weaken his position as a deal-maker in Washington.

Mr. Obama is eager to occupy the political center, Democratic officials said, to help him forge a bipartisan deal on the nation’s long-term finances that could strengthen his position heading into the 2012 election. Mr. Obama has already tacked to the center on taxes, on trade and by working to forge stronger ties with business leaders.

For their part, many union leaders worry that White House involvement could harm their case that the protests and political actions in the capitals of Wisconsin, Ohio and Indiana are grassroots, organic movements…

But others say Mr. Obama, as the leader of the Democratic Party, should do more to help the labor movement, which provides money and grassroots organizing muscle for Democratic candidates and whose power is now threatened by Republican efforts to curtail collective bargaining rights.