Some sources believe the Saints also are open to trading running back Darren Sproles.
The Saints are slammed up close to the salary cap heading into the start of free agency on Tuesday, and all three of those longtime veterans play part-time roles in New Orleans' deep offense.
Thomas and Moore have both been mentioned as possible candidates to either be released or asked to take pay cuts. Sproles would likely be even harder to pry away.
Thomas is due $2.9 million in salary and bonuses in the final year of his contract. Moore is due $3.8 million in salary and bonuses, with two years remaining on his contract.
Both players are due significant roster bonuses on March 25, with Thomas due $300,000 and Moore due $500,000, according to ESPN's Adam Caplan.
Sproles, meanwhile, is due $3.5 million in salary and bonuses in the final year of his deal.
The Saints are estimated to be just under the salary cap after factoring in a franchise-tag tender of $7.035 million to tight end Jimmy Graham. But they still need to make room to re-sign some of their own free agents and add others on the open market. They will likely carve out millions of dollars by restructuring some current contracts, as well.
New Orleans already made several tough cuts last month by parting ways with longtime veteran defensive players Will Smith, Jonathan Vilma, Roman Harper and Jabari Greer. All of those moves were expected, though, because of their diminishing roles or injury concerns.
Thomas (29), Moore (30) and Sproles (30) aren't in the exact same boat as that group; they're still at the tail end of their primes. Moreover, Thomas just had one of his best seasons to date with more than 1,000 yards from scrimmage.
None of the trio, however, played more than 50 percent of New Orleans' offensive snaps last season since the team rotates its offensive parts in and out so much.
Thomas ran for 549 yards and caught a career-high 77 passes for 513 yards and five touchdowns in 2013. He appeared in every regular-season game before suffering a chest injury that sidelined him for the playoffs.
Moore's role diminished somewhat in the Saints offense last year after he missed three games with a hand injury early in the season. He finished with 37 catches for 457 yards and two touchdowns. In 2012, he had 65 catches for a career-high 1,041 yards and six touchdowns.
News of a potential move came as a surprise, Moore said on Twitter:
It's news to you, it's news to me people. Guess we wait n see. Appreciate the love and support from everyone tho
- Lance Moore (@LanceMoore16) March 7, 2014
Sproles appeared in 15 games last season, running for 220 yards and catching 71 passes for 604 yards and four total touchdowns.