Republican Sen. John McCain of Arizona says the roll-out could happen Wednesday instead.
The bill would secure the border, remake legal immigration, boost workplace enforcement and put 11 million people here illegally on a path to citizenship.
Immigration activists and business and religious leaders had been invited to an event Tuesday to release the bill.
It would be the culmination of months of closed-door meetings among four Democrats and four Republicans, including McCain, Sen. Marco Rubio of Florida, and Democrat Chuck Schumer of New York.
Their legislation aims to strike a balance between enforcement provisions sought by Republicans, and Democratic priorities, including making citizenship accessible.
McCain, Rubio and other Republicans who wrote the immigration bill briefed other members of the Senate GOP caucus on its contents.
Afterward, some senators said they appreciated the briefing but still have much to learn before they can commit to support the legislation. Sen. Rob Portman of Ohio said there needs to be a focus on enforcement before anything else. And Sen. John Boozman of Arkansas said he has concerns about anything that could be perceived as amnesty.