ARLINGTON -- There is a week to go before next Monday's non-waiver Trade Deadline and the needle is swinging back toward Yu Darvish remaining with the Rangers.

The needle seems to swing with every trade rumor and tweet that emerges, but Major League sources indicated Monday the Rangers were strongly leaning toward not trading Darvish unless they are overwhelmed by an offer or their postseason hopes crash with another losing streak this week.

General manager Jon Daniels will have the final say and there is no doubt he has discussed Darvish with every contending team, including the Dodgers, Cubs, Yankees and Astros. Every general manager in baseball understands there is at least a remote possibility the Rangers could trade Darvish, and to not reach out to Daniels at some point would probably qualify as gross negligence. That the Dodgers have lost Clayton Kershaw for 4-6 weeks with a back injury will only increase speculation.

• Petriello: Best potential fits for Darvish

But the Rangers still have their eyes on the American League Wild Card race and those hopes were revived with the three-game sweep of the Rays on Friday. The Rangers went into Monday's game with the Marlins 2 1/2 games out in the Wild Card race. A Wild Card team has gone to the World Series in three of the last six years.

"I know we are confident we are going to make it to the postseason," Darvish said. "That's how we are playing."

Darvish met with the media briefly before Monday's game in preparation for Wednesday's start. He usually meets with the media two days ahead of a start, but he prefaced this session by making it clear he did not want to discuss trade rumors.

It has been a mentally challenging time for the right-hander beyond the normal rigors of a 162-game season. He is a free agent at the end of the season and has become a favorite topic of trade rumors as the Deadline approaches.

"I'm trying to take the same approach mentally," Darvish said. "I'm not changing anything. I'm staying strong mentally."

The Rangers have made it clear they would like to re-sign Darvish beyond this season. They also know they would receive a 2018 Draft pick as compensation if he leaves as a free agent.

The Rangers know they could get a better return in a trade for Darvish than they would if he left as a free agent. The Rangers preference would be to get young pitching in return to help offset the potential loss of Darvish, Andrew Cashner and Tyson Ross to free agency.

The question is, what kind of offer do the Rangers deem "being blown away" in a trade for Darvish? That answer may not come until next Monday, but right now the Rangers appear ready to hold on to Darvish and make a run for the Wild Card spot.

T.R. Sullivan has covered the Rangers since 1989, and for MLB.com since 2006. Follow him on Twitter @Sullivan_Ranger and listen to his podcast.