ESPN's Michael Smith first reported the fact Gonzalez was to be shopped Saturday.
A league source confirmed to ESPN that Gonzalez went to Chiefs GM Carl Peterson and inquired about a trade to a contender earlier this week.
Gonzalez, who holds the NFL tight end record for receptions, touchdown catches and yards receiving, is active in civic and charitable affairs and enjoys great popularity in Kansas City. He has always said he would like to end his career with the Chiefs, who drafted him in the first round out of California in 1997.
But the Chiefs (1-4) are rebuilding from the bottom up and not likely to contend for the Super Bowl any time soon. In spite of all his personal accomplishments, Gonzalez, 32, has never even won a postseason game.
He would like an opportunity to play in a Super Bowl and the rebuilding Chiefs would like to acquire extra draft picks.
The Giants traded tight end Jeremy Shockey to New Orleans last summer for Nos. 2 and 5 drafts pick in 2009. Shockey's replacement, Kevin Boss, has been a disappointment even though the defending Super Bowl champs are undefeated.
Michael Smith is a reporter for ESPN. The Associated Press contributed to this report.