Heckert, whose father, Tom, worked as a scout for the Browns in the 1980s, could be hired once the Eagles complete their season. They play the Dallas Cowboys in the NFC playoffs Saturday night. Heckert reportedly was going to interview for Seattle's GM job this week but backed out.
Browns president Mike Holmgren interviewed Heckert, who has been a GM since January 2006 with the Eagles, on Wednesday. Holmgren met with Seattle pro personnel director Will Lewis on Friday, but sources say the GM job is Heckert's.
NFL Network reported Friday night that Heckert likely would be announced as the Browns GM on Monday.
It's possible both Lewis and Heckert could wind up in Cleveland's front office.
During his introductory news conference, Holmgren hinted that Cleveland's hierarchy could be revamped.
"We talked about change and some of that is just restructuring," he said. "Sometimes you have five people changing the light bulb. We're going to look at everything and try and streamline things, get good people to do it and work hard."
Adam Schefter is ESPN's NFL Insider. Information from The Associated Press was used in this report.