Umenyiora is entering the final year of his contract, which is worth $3.9 million, and has long been looking for a new deal.
"I'm still open to that," Reese said during a pre-draft news conference. "There has been a little bit of talk, but it is a long way to go before we play and a lot of things for us to do. That is on our plate as well."
Also on the Super Bowl champions' plate is Victor Cruz's contract. The breakout wide receiver is also entering the final year of a contract worth $540,000. Cruz wants a new deal after setting the franchise record with 1,536 receiving yards last year.
"That is on our plate. Everything is on our plate," Reese said when asked about Cruz's contract.
Cruz, who is with the Giants during their first week of offseason training activities, reiterated that he has no plans to hold out of training camp.
"I have never been the kind of guy to hold out or rebel or anything like that," Cruz said. "I'm just happy to be here and the money and contract stuff will take care of itself. Right now you are just working out and it is OTAs and some strength and conditioning stuff.
"God willing, if I ever get to that point and get the contract that I dreamed of, it'll be an adjustment for me and probably be an emotional time for me as well. I just want to focus on being the same player I was, whether it is a $100,000 salary or $30 million or something crazy like that. No matter what it is, I just want to continue to be myself."
Umenyiora wants to continue on as a starter in addition to receiving a new contract. He told reporters last week that he still sees himself as a first-string option even though the Giants started Justin Tuck and Jason Pierre-Paul during their Super Bowl run. Umenyiora opted not to participate in this week's voluntary OTA.
"I don't like being a backup, I can tell you that," Umenyiora said last week, according to The Star-Ledger of Newark. "So, who knows how that is going to shake out. Hopefully I'll be able to stay here and finish out my career, but I just don't know. I don't know if it is going to happen."
"You can say things, you understand what I mean?" Umenyiora added about a potential extension. "I've said a lot of things. But until it happens, it's just talk, I guess, until it actually has been done."
Tuck expects his good friend to remain a Giant this season, but acknowledges he doesn't know what drama may lie ahead. Last year, Umenyiora, 30, sat out of practices during the start of camp and briefly was allowed to seek a trade. He participated in three practices after that before opting to undergo minor knee surgery.
Umenyiora never received the new contract he wanted, but had nine sacks in nine regular-season games. He also had 3½ sacks in the postseason.
"You never know with that guy and what the scenario is going to be this year," Tuck said with a chuckle. "You just wish for the best. He is a huge part of our success, especially on defense. The little things that he does in this locker room takes a lot of pressure off some of the young guys. I am definitely rooting for him to be back. You just want what is best for him."
"Until he is not a Giant, I will expect to see him," Tuck said.