Prior to the Yankees making a nine-year offer to Gerrit Cole at last week’s winter meetings, a conversation between Brian Cashman and agent Scott Boras shaped what led to a record-breaking deal for the free-agent right-hander.

“We were ready to make our first offer [for eight years] and [Boras] said, ‘Hold on.’ ” Cashman said Wednesday. “Clearly, Scott is controlling that market so he knows what is coming down the pike. He responded back to me the day before [Stephen] Strasburg [signed]. He goes, ‘Why don’t you wait in terms of engaging me until [last Tuesday]?’ There will be some new information that will be very important to you and Hal Steinbrenner to factor into whatever you are going to decide moving forward.’ ”

The Yankees GM believed Boras was talking about a development with Cole.

“[Boras] said, ‘Let me get back to you [Tuesday],’ ’’ Cashman said.

A day earlier, Strasburg agreed to a seven-year deal worth $245 million.

“I remember getting a text from Hal [Steinbrenner]. ‘Now we know what the new information [Boras] was referring to,’ ” Cashman said. “So, of course that signing affected the market and drove it even further.’’

The Yankees introduced Cole on Wednesday at a Yankee Stadium press conference during which the 29-year-old impressed with his pitching knowledge, his thirst for winning, his love of good wine, and his respect for Marvin Miller and Curt Flood. He also gave off an aura of a person very comfortable in his skin and ready to meet the challenges of working in New York.

Feeling a need to add a ninth year to the deal to keep other teams away, the Yankees offered Cole $324 million and weren’t going any higher despite a strong desire to put Cole on top of what was a solid rotation even without him.

“We had a decision to make. We felt that there was going to be a lot of [teams] in that arena of the eight-year number,’’ Cashman said. “All things being equal, West Coast versus East Coast, whoever the [teams] participating in this. Would we leave that into a close call or did we strategically feel if we went to a ninth year it would separate us from the pack. By doing so we would force them to say yes.’’

There is an avenue in which Cole’s contract would become a 10-year deal for $360 million. If Cole opts out after five years, the Yankees may void the opt out and extend the contract one more year for $36 million, The Post confirmed.

Four hours after getting the go-ahead from Steinbrenner for the record-setting nine-year contract, the deal was done. The Yankees’ second and final offer — they did make an original eight-year offer, though the terms were never publicized — according to Cashman, was the club’s best offer.

Had Boras asked for more, Cole wasn’t getting it from the Yankees.

“If it wasn’t going to be good enough it wasn’t going to be good enough,’’ Cashman said. “But there wasn’t going to be any more negotiations past that. The strategy worked. It did make the biggest difference, going to that ninth year. I believe over time the other teams would have gotten there.’’

Steinbrenner knew the deal has a good chance of pushing the Yankees past the third luxury-tax level ($248 million), but is betting Cole is a smart play.

“If you are going to do it, it needs to be a guy like this,’’ said Steinbrenner, whose team hasn’t won a World Series title since 2009 and is expecting the addition of Cole puts the Yankees in position to win multiple championships. “It’s going to be a great nine years.’’

Cole wanted the Yankees forever. He knew they would be willing to spend money and have a chance to put a World Series ring on his finger, which didn’t happen in two years as an Astro despite him going 35-10 during the regular season.

Yet, when Boras called to tell him the deal was done, Cole went silent.

“I didn’t say anything for a few seconds. He said, ‘G, are you there? You are supposed to be screaming.’ I said, ‘Yeah, I was just a little taken back.’ I was shocked. I don’t have words to really describe it. The blood kind of rushes to your gut, you get a little lightheaded and you can’t really find the words to say anything.’’