Senate Republican Whip Jon Kyl (R-AZ) comments about a lack of a 'fiscal cliff' deal while being swarmed by media off the Senate floor on Sunday. (Melina Mara/The Washington Post)

Sen. Jon Kyl (R-Ariz.) told Reuters Monday morning that a "lot of progress" has been made in talks to avert the "fiscal cliff" set to kick in at midnight Monday. 

"Conversations are still ongoing. There has been a lot of progress," Kyl said.

There isn't a deal yet, though, and asked how long conversations could go on, Kyl said, "I guess until 11:59."

Sen. Mitch McConnell’s office reported that talks continued between the Republican leader and Vice President Biden early Monday, with conversations taking place at 12:45 a.m. and 6:30 a.m. Eastern. A spokesman for McConnell expressed some hope of reaching a deal. Democrats have offered several significant concessions on taxes, including a proposal to raise rates only on earnings over $450,000 a year.