The Philadelphia Inquirer reported the deal is worth $2 million guaranteed with "a number of very-hard-to-earn incentives," according to a league source.
The 26-year-old Smith had 220 catches for 2,386 yards and 11 touchdowns in four seasons with the Giants. In his 2009 Pro Bowl season, he caught a franchise-record 107 passes for 1,220 yards and seven touchdowns.
He said Wednesday night he believes he'll be able to play soon.
"I think I'm pretty close," Smith said. "I think I can realistically be out there (the first month). It could be week one. You never know."
"I would disagree with that," Smith said. "I feel good. I've been running, cutting, doing different things. I haven't had any setbacks."
Asked why Coughlin might have indicated that Smith faced a longer rehab than Smith's doctors expressed to him, Smith said, "Maybe Coughlin had a plan up his sleeve when he said that. We'll see what happens when I get out on the field."
Smith said he believes the real reason the Giants didn't want him was financial.
"This was obviously not an easy decision for me or something I ever expected but unfortunately the business aspect proved that the Eagles wanted me to be part of their organization MUCH more than the Giants did," Smith added in a post on his Facebook page.
Smith said he couldn't speculate exactly when he'll be able to return to practice.
"I'm not sure," he said. "If it was up to me, I would say I could go practice tomorrow, but I want to be extra careful and don't want to rush it and have any setbacks. Hopefully, I'll be able to get out there really soon."
Smith enters a crowded wide receiving field with the Eagles, who also have two-time Pro Bowl pick DeSean Jackson; Jeremy Maclin, who caught 70 passes last year but hasn't practiced yet this training camp because of an undisclosed illness; Jason Avant, who caught 50 passes last year; and second-year pro Riley Cooper.
"We feel very good about our current group of receivers and Steve adds another dimension to that position," Eagles coach Andy Reid said. "He's played very well against us in the past few years and we're happy to have him on board."
Although he was almost exclusively an inside slot receiver in his four years with the Giants, Smith said he's equally comfortable in any role.
"I feel great inside, outside," he said. "The Giants used me mainly inside, but I feel like I can do both."
Smith is the sixth Pro Bowl player to join the Eagles in last two weeks, following cornerbacks Dominique Rodgers-Cromartie and Nnamdi Asomugha, defensive end Jason Babin, quarterback Vince Young and running back Ronnie Brown.
"(General Manager) Howie Roseman and I are always keeping our eyes open for good football players and players that we think can help our football team win," Reid said. "And Steve Smith certainly fits that category. He's a Pro Bowl-quality receiver that we will work into our offense as soon as he is ready to go."
The Associated Press contributed to this report