The NL championship series MVP agreed to a $20 million, three-year contract with the Giants late Tuesday. Vice president of baseball operations Bobby Evans said only a physical was left to finalize the deal.

“He was a priority from Day One,” Evans said. “He was a key part of our 2012 success and served as a tremendous veteran presence.”

Earlier in the evening, Scutaro had been weighing a two-year contract offer that included a vesting option — but he was seeking a three-year deal to remain with the World Series champions. The Giants and Scutaro’srepresentatives met Tuesday at the winter meetings in Nashville, Tenn.

In other moves, the Giants exercised 2014 contract options on manager Bruce Bochy and GM Brian Sabean.

The 37-year-old Scutaro batted .362 with three homers and 44 RBIs in 61 regular-season games with the Giants after he was acquired in a July 27 trade with Colorado. He frequently said how much he enjoyed playing for San Francisco and that he hoped he would return.

The Giants suddenly have quite a familiar roster returning after making two big moves in as many days to keep their own free agents. On Monday, center fielder Angel Pagan agreed to a $40 million, four-year contract.

Left-handed reliever Jeremy Affeldt received an $18 million, three-year contract on Nov. 14.

Retaining Affeldt, Pagan and Scutaro were among the top priorities for Sabean and Evans this offseason.

Giants CEO Larry Baer considered it a productive winter meetings — and offseason so far — for his busy club after already re-signing Affeldt last month. Winning the World Series for the second time in three seasons put San Francisco behind on its offseason plan again, though everyone in the front office would say that’s a great problem to have.

“Culture is important, and Marco is a wonderful influence with our players, especially the Latin players,” Baer said.

The 31-year-old Pagan batted .288 with eight home runs, 56 RBIs and a San Francisco-best 15 triples in his first season with the Giants. Pagan said late in the season and again after the World Series parade that he hoped to return, but wanted to test free agency and sought some job security in the form of a multiyear deal.

On Friday, Sabean said the club had made backup plans to move forward without Pagan or Scutaro in case neither decided to re-sign — but progress with Pagan apparently happened during the weekend after Sabean said, “We don’t have a deal, so that tells you we’re not close.”

San Francisco declined Friday to tender a contract to closer Brian Wilson, the 2010 major league saves leader with 48. The right-hander is working his way back from a second Tommy John surgery on his pitching elbow.