Umenyiora has now been offered a range of $400,000 to $1 million in new money, based on incentives and production.
Umenyiora has not accepted any restructured deal and prefers to be traded at this point, according to sources.
Umenyiora went to Georgia for a second opinion and is likely to practice Monday for the first time this training camp. He has been seeking a new deal more in line with his production.
The Giants have offered deals with incentives based on being a top-10 defense in several categories, as well as based on playing time. Umenyiora set an NFL record for forced fumbles last season and is set to make under $8 million total on the final two years of his deal.
Baltimore, Seattle, San Diego and New England are among the teams interested in dealing for him -- the Seahawks had been mulling trading tight end John Carlson and a third-round pick for him, according to league sources -- but the Giants had held out for a first-round pick.
Follow Jason La Canfora on Twitter @jasonlacanfora.