Less than 10 days now remain until the Aug. 1 non-waiver Trade Deadline, and talks are shifting into high gear as teams search for stretch-run upgrades.
While no major deals went down on Saturday, there was no shortage of rumors to be found. Here is a look at the latest:
Rays aiming high
Rumors have swirled about the Rays trading one or more members of their starting rotation, a group that includes Matt Moore and Jake Odorizzi.
One team with reported interest in those two is the Rangers, but according to MLB.com's Jon Paul Morosi, the two teams are engaged in a "staring contest." While the Rays want talented 23-year-old infielder Jurickson Profar in exchange for one of those pitchers, the Rangers so far are declining to part with him in such a deal, Morosi reported. Several Texas prospects also have been part of the discussions.
Meanwhile, Tampa Bay believes that about eight teams have enough talent in their farm systems to pull off a deal for fellow starter Chris Archer, according to Ken Rosenthal of FOXSports.com. Archer has struggled at times in 2016, posting a 4.60 ERA overall, but he has a strong track record, is 27 years old and is signed to a club-friendly contract.

Latest on Sale
Moore and Odorizzi, along with the Padres' Andrew Cashner, appear to be part of the Rangers' backup plan if they are unable to land White Sox left-hander Chris Sale, according to MLB.com's T.R. Sullivan. And Texas needs alternatives, because Sale won't come cheap, despite the pregame clubhouse incident that led to him being scratched from his start on Saturday.
In fact, Chicago reportedly wants five top prospects for Sale. Sullivan reported that while Texas is "being flexible" when it comes to Profar, Joey Gallo and others, the club is not willing to part with young outfielder Nomar Mazara -- even for Sale.
An unnamed general manager told Jeff Passan of Yahoo! Sports that the Sox are asking for a "staggering" return package, although two other executives involved in talks with the club told Passan they think a Sale deal will happen. The Sox can shoot for the moon because Sale (14-3, 3.18 ERA) has a 2.99 ERA in 135 career starts and is set to make $38 million over the next three seasons, if two club options are exercised.

Chapman on the move?
The Yankees are informing other clubs that they are nearing a trade of closer Aroldis Chapman but will hang on to fellow left-handed reliever Andrew Miller, Rosenthal reported on Saturday night. Rosenthal's report didn't name the second team in a potential deal, but MLB.com's Bill Ladson wrote on Friday that Chapman was the Nationals' top target. 
Washington also is among the teams that have "checked in on" Royals closer Wade Davis, according to Jon Heyman of Today's Knuckleball, who added that the asking price for him appears to be higher than Chapman's. While it's not yet clear that the Royals will sell, Davis would be an attractive chip. He owns a 1.00 ERA over the last three seasons, and his contract includes a $10 million club option for 2017.

Avila: No big moves from Tigers
Tigers general manager Al Avila said Saturday that he does not expect the Tigers to do anything drastic before the Deadline, whether that's adding or subtracting. Part of the reason is that Detroit is scheduled to get starters Jordan Zimmermann and Daniel Norris back from the disabled list as early as next week, with right fielder J.D. Martinez perhaps returning in early August.
Considering the high asking prices Avila has encountered in trade talks, especially for pitching, he prefers to stick with internal additions for a club that entered Saturday 2 1/2 games out of the second American League Wild Card spot.
"This is a team that can get us into the playoffs, and we just have to get back to guys that are injured," Avila said.