The 25-year-old is set to become a free agent at the end of the season, and was non-committal when questioned by the media after the Giants' Divisional Round loss to the Philadelphia Eagles on Saturday.
Jones was sacked five times in Philadelphia, but had a solid regular season, throwing 15 touchdowns and rushing for another seven in 16 games, while overall he completed a career-best 67.2 per cent of his pass attempts (317 out of 472).
Regarding his contract, Jones said on Saturday: "We will cross that bridge when we get there. I really enjoyed playing here... We will figure out where that goes, but I have nothing but love and respect for this organisation and the group of guys in the locker room."
Speaking at an end-of-season press conference on Monday, Schoen said: "We'd like Daniel to be here. He said it, there is a business side to it. But we feel like Daniel played well this season. He's done everything we've asked him to... We would like to have Daniel Jones back.
"Hopefully we can get something done with his representatives. That would be the goal, to build a team around him where he could lead us to win a Super Bowl."
Having turned down the chance to exercise the fifth-year option on Jones' rookie contract, it seems the Giants feel he has proven himself worthy and Schoen is eager to get to the negotiating table.
"It takes two. Both sides are going to have that conversation," he said. "We haven't crossed that bridge yet. There are tools at our disposal."