Negotiations continued Tuesday between the New York Giants and representatives for injured defensive end Jason Pierre-Paul, as the two sides discussed a one-year, incentive-heavy deal that would get Pierre-Paul into the team facility this week and possibly playing by Week 10 or Week 12, according to sources familiar with the talks.

There's greater optimism from the Giants' side than there was when he visited last month about Pierre-Paul's ability to return and play this season. Team doctors examined him Monday, and the fact that the sides were talking contract after that is an indication that he could be on the verge of a return to the team. But there remains a chance the negotiations don't end up producing a deal, in which case the Giants likely would rescind Pierre-Paul's $14.813 million franchise tender and make him an unrestricted free agent.

Along with his hand injury, the Giants remain concerned about Jason Pierre-Paul's conditioning. Brad Penner/USA TODAY Sports

ESPN's Adam Schefter reported Monday night that Pierre-Paul met Monday with Giants GM Jerry Reese, coach Tom Coughlin and defensive line coach Robert Nunn, and that the team told Pierre-Paul it wanted him back. That's a strong positive sign for Pierre-Paul's chances of actually coming back, because the last time he was in town, the Giants told him they didn't think he was anywhere near ready and to return in six weeks for another checkup on the right hand he severely damaged in a July 4 fireworks accident.

That was seven weeks ago, and Pierre-Paul returned Monday for another exam on his hand. Team doctors have been examining him, as has hand specialist Robert Hotchkiss from the Hospital for Special Surgery, who examined him last month as well. The Giants are using the results of the medical exam to determine what they can reasonably expect from Pierre-Paul once he does return. It's not just the condition of the hand that concerns them; there are also legitimate questions about what kind of conditioning Pierre-Paul has or has not been able to do since the accident.

Pierre-Paul has shown the Giants video of his working out with a private defensive line coach he hired to help refine his technique, and he's posted videos of himself lifting weights on his Instagram account. But the Giants have taken the stance that they'll base their evaluation on what their own doctors and trainers think about Pierre-Paul's physical condition, and Monday's visit was part of that process.

The medical information is necessarily tied into the financial discussion, since the amount of money the Giants are willing to pay Pierre-Paul this year is tied to the kind of production they think they can expect. The team wants Pierre-Paul to accept a deal that's far lower than the $14.813 million franchise tender it offered him in March, though the Giants are willing to offer him the chance to make some of the money back through incentives. They're not allowed to sign him to a deal that extends beyond this year, because the franchise player rules prohibit teams from signing their own franchise players to long-term deals after July 15.

There's a sense among the people I've talked to that the situation could be resolved one way or the other Tuesday. There are a lot of potential outcomes still on the table, but a resolution is likely to happen soon. Pierre-Paul's season officially ends if he doesn't sign his franchise tender by Nov. 17, and Monday was his second official free-agent visit, which is the most allowed under NFL rules. The Giants can't work him out or examine him again without signing him first.

The Giants play at New Orleans, at Tampa Bay and home against New England the next three weeks and have a bye in Week 11. Assuming he signs this week, the timing of Pierre-Paul's potential return to the field would depend on how quickly he progresses in his rehab and conditioning over the coming weeks.