Such speculation hasn't hampered his ability to be successful, with Vargas going 4-0 in the month with a 1.46 ERA in five starts.
"I hope that what I can do out on the field is worth them keeping me here," Vargas said. "Obviously it means you're doing all right if they're talking about other teams wanting you, but this is where I want to stay. This is a great city, I love the organization, I love my teammates, I love the staff -- I can't say any more than that. It's home to me."
On Thursday against the Royals, he was superb, allowing just one run on one hit in eight innings of work. Vargas also struck out five to improve to 11-7 on the season with a 3.76 ERA.
For manager Eric Wedge, the start showed once again how much Vargas has grown since coming to the Mariners more than three years ago.
"He's just grown. Even the short period of time I've been here, it's been a lot of fun to watch this guy continue to figure it out, continue to get better. I think that when you talk about the way he handles different situations, every game's different, the way he reacts to different situations, executing pitches at different points in time when it's really important," Wedge said. "His execution, his mindset, his understanding of what he needs to do to be successful. It's been a lot of fun to watch him grow."
Meanwhile, Aaron Laffey will be on the hill for the Blue Jays. The left-hander took a no-decision his last time out after winning two straight starts.
He allowed four runs on five hits in 5 2/3 innings against Oakland on Thursday, tossing a season-high 107 pitches.
"There were a lot more than a few mistakes," Laffey said after the start. "I walked three guys, put myself in a bad position. I just fell behind some guys and the defense picked me up big time."
Blue Jays: Bautista to face live pitching
The Blue Jays hope to speed up Jose Bautista's return to the lineup by having him remain with the club and take swings in a simulated-game environment rather than being sent to the Minor Leagues.
Bautista, who is currently on the 15-day disabled list with a sprained left wrist, has begun taking swings but has yet to face live pitching.
"We'll see how aggressive and intense we can create some of the live BP here first," Blue Jays manager John Farrell said. "Any available arm that can throw decently, move the cage up, whatever, we'll figure out those ways first. Jose, as honest as he is, he'll indicate to us what he feels most comfortable with.
"Two weeks is a long time without facing live pitching. It has been done before but we'll have to cross a few bridges here before we get to that point."
Mariners: Gutierrez undergoes more tests
Center fielder Franklin Gutierrez underwent more medical tests on Monday as the Mariners continue to monitor his progress from a concussion he received June 29 after being hit in the head by a pickoff throw.
Gutierrez remains on the 15-day disabled list, but has been doing well while running, shagging fly balls and taking some batting practice the past few days, manager Eric Wedge said.
"We're just going to keep eliminating and pushing forward and keeping a close eye on him," Wedge said. "Hopefully everything comes back good today and we can keep him going. But he had a good day yesterday with all the activity, which was a good sign."
Blue Jays closer Casey Janssen is 13-for-13 in save situations since assuming the closer's role on May 9. In those 13 saves, he's allowed just six baserunners in 13 2/3 innings.
The Mariners have held leads in 27 of their 57 losses, which leads the American League.
Over his past 39 games, Mariners designated hitter John Jaso is hitting .316 (31-for-98) with four home runs, 20 RBIs and 15 runs scored.