According to a Republican official, Mr Bush is expected to announce the choice as early as Wednesday.
Mr McClellan announced his resignation last week as part of a staff shake-up engineered by new White House Chief of Staff Josh Bolten in a bid to revive Mr Bush's presidency.
The 50-year-old former speechwriter for former President George Bush, Mr Snow has been host of Fox News Radio's The Tony Snow Show.
For a time he was the anchor of the Fox News Sunday program and was treated for colon cancer last year.
The Washington Post said Mr Snow had accepted the offer after top officials assured him he would be an active participant in administration policy debates, not just a spokesman.
The Post quoted sources as saying Mr Snow viewed himself as well-positioned to ease the tensions between the Bush White House and the press corps because he understands politics and journalism.
Mr Snow was said to have wrestled with the decision for several days on whether to take the job.