Rodriguez is in Miami to meet with Dr. Lee Kaplan on Sunday afternoon. Results of the second opinion are expected to be available on Sunday evening, according to manager Joe Girardi.
An MRI on Friday revealed the slight tear. He is expected to decide on whether or not to have surgery following the meeting with Kaplan.
If the 35-year-old opts to undergo surgery, he would miss approximately one month.
Girardi said on Sunday that, if Rodriguez opted for surgery, undergoing the procedure now would be ideal given the Yankees will have four days off during the All-Star break.
"If you're going to have the surgery let's get it done as soon as possible," Girardi said.
The manager expressed concern that Rodriguez would be forced to play eight consecutive games on turf after the All-Star break if he decides to forgo surgery. The Yankees open the second half with a four-game series in Toronto and then head to Tampa Bay for a four-game set.
"I don't know if that's going to be any easier; my guess is it's going to be a little bit harder," Girardi said. "It's difficult but it's a choice that he has to make."
Rodriguez originally suffered the injury on June 19 in Chicago. He thought about scoring on a passed ball, but ultimately decided against it. And in the process of turning back toward third base, Rodriguez tweaked his knee.
"I want to talk to (Yankees team physician Dr. Christopher) Ahmad and then I'm gonna see Dr. K and we'll call Dr. Ahmad and make a decision collectively," Rodriguez said after not playing in the Yankees' 5-4 win over the Rays on Saturday afternoon at Yankee Stadium. "It's hard to get into missing 20, 30, 40 games when you don't know, but at the same time you don't want to limp toward the finish line. The most important games are played at the end."
He has already withdrawn from the 2011 All-Star Game on Tuesday in Phoenix.
"It is what it is," Rodriguez said. "Most players play with bumps and bruises, but you never expect it to be a tear. Obviously, it was disappointing."
Asked about returning if he does elect to go under the knife, Rodriguez said, "It's been done before."
If he doesn't opt for surgery, Rodriguez could try to play through it and have the procedure in the offseason.
Girardi said Saturday that pitcher CC Sabathia played through the same injury last season.
"It was tough but it's different for me because I cover first maybe once every other game and things like that and he's out there diving for balls and trying to drive balls and running the bases," Sabathia said. "I can only imagine how tough is it."
Girardi said Saturday the Yankees would deal with whatever choice Rodriguez makes.
"Players have done it before. You're not ever sure how productive they're going to be and if you're going to need to get them a day off here. It's unpredictable," Girardi said. "If he wants to try it, that's something we'll definitely discuss. But if he decides to have surgery, we'll live with that too."
Outfielder Andruw Jones also suffered from the injury during 2008, when he played for the Dodgers. He woke up one day and said he couldn't walk and opted to have the surgery. The injury caused a leaking of fluid in his knee and a cyst on the back of his knee had to be drained.
As he tried to come back, Jones said he was running in his second week and came back in four weeks, but that was too soon and he got hurt again and landed back on the disabled list. Jones added that it affected his power and he did not have his leg under him at the plate.
Rodriguez has not hit a home run in his past 85 at-bats, the longest drought of his career, and has just seven extra-base hits in that stretch.
Rodriguez, who is batting .295 with 52 RBIs in 80 games, has just 13 home runs so far, putting him on pace for 24 for the season. That would snap his streak of 13 straight seasons with 30 or more home runs.
"I just don't think that he has the drive in his backside that he needs to have to be the power hitter that he's capable of being," Girardi said Saturday. "I still think he can productive and get his base hits but I don't think you're going to necessarily see him drive the ball as much."
Girardi said he had watched Rodriguez in the past few games, including an 0-for-4 night against the Rays on Thursday, and when he decided to take him out of Friday's lineup, he broached the topic of Rodriguez's having an MRI.
Girardi recognized that the Yankees won't have Rodriguez at 100 percent if surgery is postponed, and he'll have to give him more time off in the second half of the season.
If Rodriguez does have the surgery, Eduardo Nunez would step in at third base. Nunez is hitting .278 with three home runs and 14 RBIs this season.
Girardi said there is no timetable on the return of Eric Chavez, a natural third baseman, who has been out since May 6 with a fractured fifth metatarsal in his left foot and now is dealing with back issues as well.
"I want to talk to Alex and see how he feels and what he thinks," Girardi said. "And then form my opinion on it. I don't want to influence his decision because I don't know how he truly feels. That's only something he can truly evaluate so I want to listen to him and discuss it with him."
Mike Mazzeo and Matt Ehalt are regular contributors to ESPNNewYork.com. Information from ESPNNewYork.com's Ian Begley and Wallace Matthews was used in this report.