The Denver Post reported Hancock wore the jersey during halftime of the Nuggets' home game against the Utah Jazz Sunday night.
Denver lost Saturday's playoff game 45-10 to New England.
Hancock also is sending Menino 40 pounds of Colorado beef, including some "Tebow tenderloins."
"Well, I think Tim's earned the right to be the starting quarterback going into training camp next year," Elway said.
The Associated Press contributed to this report.