"Our first choice with respect to Osi is that he will play for the Giants and retire a Giant," Reese said. "That is what we would like to happen. Osi has been offered an extension two years in a row now, so we would still like to make it work. Hopefully, it will work out."
Umenyiora said earlier this week that he did not think the Giants would be able to give him the type of extension he is seeking with several young players due for raises in the future. The defensive end long has wanted a new contract and is entering the final year of his deal worth just under $4 million. Umenyiora, 30, said the Giants offered to add incentives to his contract last year but he declined.
Umenyiora wants to remain a Giant but not under his current contract. He suggested before the draft that the Giants should trade him to get something in return in the draft if they don't plan on giving him the kind of extension he is seeking.
When asked if the Giants explored any trades for Umenyiora, Reese replied, "All options are open with respect to that."
The Giants recently signed defensive end Mathias Kiwanuka to a three-year contract extension through 2015. The extension is worth $21.75 million, according to reports. Reese said the extension had nothing to do with Umenyiora's situation.
"Happy for Kiwi; he deserves it for sure," Umenyiora wrote in an email on Friday in response to Kiwanuka's extension. "Especially all he has been through. A very good player. Honestly, it's a good, smart business move by the Giants. Take care of a good player and buy some insurance at the same time. Very well done.
"How that impacts me remains to be seen, though," Umenyiora added. "But I understand why they would do that (extension for Kiwanuka). It makes sense."
The Giants did not select a defensive end in the draft. Umenyiora said he is unsure what he will do if the Giants do not give him a new contract by training camp. He sat out practices during the first two weeks of last year's camp before eventually opting for minor knee surgery.
"Would I want to (stay)? I think it would be ideal," Umenyiora said on Wednesday while promoting the NFL Play 60 Youth Football game for Xbox 360 Kinect. "I think I would if things could be worked out. But is that going to happen? I really don't think so.
"I doubt it because of the situation that they have with Jason (Pierre-Paul), Justin (Tuck), Victor (Cruz) and Hakeem (Nicks) coming up (eventually)," Umenyiora added. "I feel like I'm one of the very good pass-rushers in the NFL; I think my statistics have proven that. Are they going to be able to pay me as such? I seriously doubt that."