When his phone rang Friday and Jeff Gorton saw the Boston exchange and the caller ID, he answered right away.

After Jimmy Vesey — the standout Harvard free-agent forward who had been wooed by eight NHL teams this week — told Gorton he wanted to come to New York, the Rangers’ general manager said his joyous response was not the sort to be repeated for public consumption.

“After our meeting with him on Wednesday, I felt we made a connection,” Gorton said last night. “But until he called me, I was uncomfortable. You never know with these types of situations.”

Indeed, Gorton’s comfort level of this retooling franchise rose with Vesey, a 23-year-old left wing who had 56 goals and 104 points in 70 games as a junior and senior. He agreed to a two-year contract and chose the Blueshirts over his hometown Bruins, the Chicago Blackhawks, the Toronto Maple Leafs and the Islanders, among others. “We’re trying to get as many young players as we can,” Gorton said.

“It was definitely a tough decision,” said Vesey, who was “very impressed” by the pitch from Gorton, coach Alain Vigneault and director of player development Chris Drury, who, like Vesey, had won the Hobey Baker Memorial Award as the nation’s top college player. “It jumped out they really want me . . . It seemed that they were going to have me in their lineup and believed in me.”

For a team without a first-round draft pick for the last four years, Vesey — who was drafted 66th overall by Nashville in 2012 but never signed and chose free agency Aug. 15 — is a no-risk win. Gorton didn’t surrender anyone in a trade or pay a fortune. Under the league’s collective-bargaining agreement, he will receive an annual $925,000 entry-level salary and up to about $1.9 million in performance bonuses.

The wooing of Vesey, a 6-2, 205-pounder who projects to play on the second or third lines, included conversations with stars from other teams such as John Tavares and Patrick Kane and tweets from New York celebrities and athletes encouraging him to come to Manhattan.

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He didn’t plan that. “I’m a quiet kid,” he said. “I had no intention of it being such a media circus; I’m glad the process is over. I put pressure on myself on my own.”

But in talking to the Rangers contingent, Vesey said he understood that the front office “doesn’t really expect me to come in as a savior. I’m a piece that they want to add.”

“We have a lot to sell,” Gorton said. “An Original Six team, a lot of good players, we try to win. The fact that he’s from the East I think helped a little bit.”

Vesey had a sense of what it would be like to skate for the Blueshirts, having played at Madison Square Garden with Harvard three times and practicing at the Rangers’ training complex in Westchester County.

He also said that talking to longtime pal Kevin Hayes, a Rangers center, “had a big impact.”

Coming out of college to the pros will be a major adjustment, but Vesey, considered a two-way player with a scoring touch and a top-shelf work ethic, understands the task. “I feel New York is somewhere I can plug in right away,’’ he said, “but I’m going to have to put in the work.”