Defensive end Jason Pierre-Paul of the Giants warms up before...

Defensive end Jason Pierre-Paul of the Giants warms up before a game against the Arizona Cardinals at MetLife Stadium on Sept. 14, 2014 in East Rutherford, N.J. Credit: Getty Images / Ron Antonelli

The Giants remain cautiously optimistic that defensive end Jason Pierre-Paul, who severely injured his right hand in a July 4 fireworks accident, can return before the end of the season, possibly for the final four games or even longer, according to a league source familiar with the team's situation.

Pierre-Paul's injuries resulted in the amputation of his index finger and portions of his thumb and middle finger. The source said the damage to the middle finger is more extensive than previously thought, although the Giants think it wouldn't prevent Pierre-Paul from being able to play as long as skin grafts used to repair his hand and fingers are strong enough.

"If he gets back for four games, that would be good," the source said. "If it's more, that would be a bonus."

The source added, however, that the team remains realistic about the possibility that a return might not happen this year if further healing doesn't take place. "It's still possible he doesn't come back [this year],'' the source said.

Pierre-Paul has been designated as the Giants' franchise player, and he has until Nov. 17 -- after the 10th game of the season -- to sign a contract. If he does not sign by then, he is prohibited from playing for the remainder of the season, according to rules governing the franchise tender.

Pierre-Paul visited with the team Sept. 7 to be examined by its medical staff. He returned home to South Florida later that day and was told by the Giants to continue rehabilitation exercises while he heals.

According to NFL policy, a free agent can work out for a team twice before agreeing to a contract, which means the Giants could have one more meeting with Pierre-Paul before determining a course of action. The team prefers to wait until Pierre-Paul is closer to being able to play before meeting with him for a second time.

If Pierre-Paul does play this season, the source said he could decide to tape his middle finger, ring finger and pinky together to make up for any weakness caused by the damage to his middle finger.

The Giants have not pulled the franchise tender, which was worth $14.8 million -- or about $871,000 per game -- before the start of the season. If Pierre-Paul does return, he could agree on a contract tied to his availability to play in games.

If the Giants do re-sign him, they will be granted a two-week roster exemption before placing him on the active roster or putting him on the non-football injury list. If he is put on NFI, the team is not obligated to pay him, although the Giants could structure a contract in which he would be paid even if he didn't play.

The Giants' defense has done reasonably well in Pierre-Paul's absence, but the pass rush still is having problems. The Giants are tied for 27th with only five sacks in four games. Lineman Cullen Jenkins leads the team with two sacks.

The Giants are 25th in yards allowed (386 per game) but are a respectable 11th in points allowed (20.5).

Many of Pierre-Paul's teammates are holding out hope that he can return, and Pierre-Paul has indicated through social media posts that he is getting closer to being ready to play.

In a Sept. 15 Instagram, he posted a picture of himself working out and wrote, "God can turn any situation around for the better. Your hard times will lead to the best times of your life. Don't believe the hype . . . Be back soon!"

Cornerback Prince Amukamara was asked after the Giants' 24-10 win over the Bills on Sunday if he thinks Pierre-Paul will play this season.

"I would hope so," Amukamara said. "I pray [he will return] every night."

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