The Rangers can now trade any of these players and considering their huge contracts an opposing team would have to take a substantial amount of the remaining years of the contract to make a deal happen.
The Rangers are 48-76 this season.
Team officials don't comment on players clearing waivers, however, the moves are interesting with the Rangers looking for high-quality prospects.
Two weeks ago, the team was able to get starting right fielder Alex Rios through waivers, but he has yet to be traded.
Reliever Neal Cotts, another player who could be sent through waivers, has interest from several teams.
The biggest name on this list is Darvish, who is on the 15-day disabled list with inflammation in his right elbow. Darvish is eligible to return Aug. 25 at Seattle, and he most likely would have to throw a pain-free bullpen session to pitch in that game.
On the season, Darvish is 10-7 with a 3.06 ERA. But he struggled in his last start, Aug. 9 at Houston, giving up six runs -- five earned -- in four innings of work during an 8-3 loss.
The club signed Darvish to a six-year, $56 million deal in 2012.
Andrus and Rios are two realistic possibilities for getting traded.
Rios enters the final year of his contract, but the club has a $13.5 million club option on him for 2015. Rios and the club expressed interest in him returning next season. But if some valuable prospects can be obtained, Rangers general manager Jon Daniels would likely make a move.
Rios is battling a sprained right ankle and some reduced power numbers. He has just four home runs on the season with 49 RBIs. His on base percentage is .313, below his career average of .323.
The organization doesn't believe Odor can play shortstop meaning Profar, who has missed this season due to shoulder surgery, could get a look at that position if Andrus wasn't on the team.
Luis Sardinas (Triple-A Round Rock) and Travis Demeritte (Single-A Hickory) are two highly regarded prospects in the farm system who are projected to be one or two years away from being ready for the big leagues.
Andrus' new eight-year, $120 million deal begins in 2015 where he'll get a base salary of $15 million for next season.
Finding a trade partner willing to take a load of that salary, if not all of it, will be hard. Andrus has struggled at times this season. His on base percentage is .332, the lowest of his six-years in the big leagues. He's also scored 61 times and produced 132 hits.
Andrus has hit into 17 double plays -- just two shy of his career-high 19 he hit into last season.
Andrus is still a valuable player for the Rangers, and he's taken on more of leadership role given the difficult season and the number of younger players around him.