Markakis and the Orioles have made "real progress" in their contract-extension negotiations in recent days, according to club president Andy MacPhail.
"We're close," MacPhail said, without revealing specifics. "But we're not there yet."
The source said that once some minor details are worked out, a deal could be announced this week, assuming the right fielder passes a routine physical in Baltimore.
The goal is for the agreement to be reached officially by Tuesday, when agents and ballclubs must report arbitration figures to the commissioner's office. Markakis is in his first year of arbitration, and both sides want to avoid the potential acrimony that often comes with that process.
The expected deal would keep Markakis, 25, with the Orioles through the 2014 season. Currently, he is eligible for free agency after 2011.
Depending on the final value, the contract could be the second-most expensive in Orioles history, behind only the six-year, $72 million deal given to shortstop Miguel Tejada, the 2002 American League Most Valuable Player, in December 2003. In 1998, the Orioles signed outfielder Albert Belle to a five-year contract worth $65 million.
Markakis would have been looking for a big raise in arbitration after making $455,000 in 2008, when the Orioles unilaterally renewed his contract. The sides engaged in discussions at the All-Star break, but Murphy turned down the Orioles' offer that was believed to be worth between $5 million and $6 million annually.
MacPhail said at the start of the offseason that signing Markakis to a long-term deal was one of his priorities. In December, the Orioles offered a six-year, $60 million deal and Murphy countered, asking for roughly $10 million more. The sides broke off negotiations just before baseball's winter meetings and then resumed talks this month.
It appears they have met in the middle - which would allow the club to keep its top building block for half of the next decade while providing security for Markakis, who got married last year and owns a house in Baltimore.
His wife is due in March with the couple's first child.
The Orioles drafted Markakis, who was a star outfielder and pitcher at Young Harris Junior College in Georgia, with the seventh overall pick in 2003. Though several teams believed that he had more value as a left-handed pitcher, the Orioles were enamored of his sweet, powerful swing and patient approach.
He played 279 games in the minor leagues before his torrid spring training performance in 2006 earned him a spot on the team's Opening Day roster. After a slow start to his rookie season - he was hitting .219 at the end of May - Markakis blossomed, finishing the year with a .291 average, 16 homers and 62 RBIs.
He had an even better year in 2007, batting .300 with 23 homers, 112 RBIs and 18 stolen bases while establishing himself as one of the best young players in the game.
Frequently pitched around last season, Markakis still managed to hit .306 with 20 home runs and 87 RBIs.
He set career highs in average, runs (106), doubles (48), walks (99) and on-base (.406) and slugging percentage (.491).
He also led the major leagues with 17 outfield assists and was widely considered a Gold Glove snub.
Baltimore Sun reporter Jeff Zrebiec contributed to this article.