The Giants have scored the fewest runs in the majors this season. They needed a balk to beat the Cleveland Indians, 1-0, on Saturday afternoon. They could upgrade their lineup in any number of ways.

Right now, though, they remain focused on acquiring a catcher, major-league sources told FOXSports.com.

With Buster Posey out for the season, the Giants are using a tandem of Eli Whiteside, 31, and Chris Stewart, 29. Neither is hitting particularly well: Whiteside .221, Stewart .156.

Sources say the Giants have shown interest in Texas catcher Yorvit Torrealba, who began his career in San Francisco and has been the Rangers’ primary backstop this year.

But there are two significant roadblocks to a deal: The Rangers can’t seriously consider trading Torrealba until fellow catcher Mike Napoli comes off the disabled list; and the Giants are reluctant to pay the $3.25 million on Torrealba’s contract for next season, when Posey will presumably be back behind the plate.

Texas would want a bullpen arm in return, a source indicated, and Giants general manager Brian Sabean is said to be reluctant to part with any of his key late-inning relievers. But veteran right-hander Guillermo Mota, with a 3.89 ERA in 37 innings, is available in the right deal.

The Giants have discussed a number of other catchers internally, sources say, including the Nationals’ Pudge Rodriguez, the Mets’ Ronny Paulino, and the Reds’ Ramon Hernandez. It’s not known if they have made progress on a deal involving any of them.

Hernandez is an intriguing possibility. This year, Ryan Hanigan has started more games than Hernandez behind the plate in Cincinnati. Meanwhile, Reds catching prospect Devin Mesoraco could become a frontline major-league catcher in the near future. Mesoraco, who just turned 23, has a .936 OPS at Class AAA Louisville this year — including a scalding 1.035 this month.

If Mesoraco’s first call-up is drawing near, the Reds could become more willing to listen to offers for Hernandez. But Cincinnati would probably need to address one of its biggest needs — starting pitching or shortstop — in order to move him. (And the Giants have been looking for a shortstop themselves, although that’s a lower priority than catching for them.)

Due to the lack of production elsewhere, San Francisco appears willing to sacrifice defense for offense at catcher more than a National League team normally would. That is particularly true in light of the fact that Whiteside has thrown out only 20 percent of potential base-stealers this year. (Torrealba, by comparison, is at 33 percent.)

The Giants have one of the best pitching staffs in baseball, but their depth was compromised with the news Saturday that left-handed starter Jonathan Sanchez was placed on the disabled list with left biceps tendinitis. Barry Zito came off the DL to replace Sanchez in the rotation.

-- Jon Paul Morosi and Ken Rosenthal