Amid all the shuffling in latenight during the past few months, David Letterman has quietly been making plans to stay put at CBS for at least three more years.

It’s understood CBS and Letterman are deep in discussions on a new deal that would keep him at the helm of “Late Show With David Letterman” through the 2011-12 season. Letterman’s existing deal with the Eye runs through the 2009-10 season, but the new three-year pact under discussion would supersede the final year of the previous contract.

That’s significant for the Eye because the network is said to be negotiating a reduction in the license fee that it pays for “Late Show.” CBS has been on a cost-cutting binge on all of its programs, so it’s no surprise that “Late Show” would also be facing a license fee reduction.

Letterman’s Worldwide Pants shingle owns “Late Show” outright, along with its 12:35 a.m. companion, “The Late, Late Show With Craig Ferguson.” Letterman’s personal paycheck has long been said to be one of the richest in TV, pegged at about $30 million annually for most of the past decade. It’s unclear how the license fee reduction will affect Letterman’s take-home pay.

CBS and a rep for Worldwide Pants declined comment late Tuesday.