The Twins are entertaining offers for left-hander Johan Santana, the Athletics are dangling right-hander Dan Haren and the Orioles are listening to offers for left-hander Erik Bedard.
There were rumors circulating through the Gaylord Opryland Hotel & Resort on Wednesday afternoon that the Mariners were back in the Johan Santana sweepstakes.
"In two significant ones [Santana and Haren], I would say we're not in play and the other one we are," Bavasi said during his nightly briefing with Seattle-area reporters.
As of Wednesday night, Seattle remained among the teams that could have a chance of acquiring left-hander Erik Bedard from the Orioles. The Mariners are believed to have made an offer that piqued Baltimore's interest, but the AL East team wants to explore other options before seriously considering Seattle's proposal.
Also, the Birds want left-handed reliever George Sherrill in any deal made with Seattle.
"The premier pitching that's on the market, I would doubt, has a chance of getting moved without getting some quality pitching back," Bavasi said. "They would want quality and numbers."
The Mets reportedly sent the Orioles an offer to trade right-handers Aaron Heilman and Philip Humber and outfielder Carlos Gomez for Bedard. That offer was quickly declined.
The Dodgers also have shown interest in Bedard, who had a 13-5 record and 3.16 ERA last season.
"I think we've said it every way we can," Orioles president of baseball operations Andy MacPhail said. "It's not a question of really shopping [Bedard] or trying to move him. But at the same time, our record is what our record's been. We have a responsibility to our fans to try to make it better and take whatever we think we have to take to make it better."
Bavasi said the Mariners are still talking to clubs about trades and also have increased offers to at least two unidentified free agents. It is being speculated that free-agent right-hander Hiroki Kuroda now has a four-year offer worth about $40 million on the table.
That's one year more than either the D-backs or Dodgers have offered, but the D-Backs have increased their pursuit of Kuroda. The organization put together a video that includes messages from CEO Jeff Moorad, president Derrick Hall, GM Josh Byrnes and left-hander Randy Johnson, who filmed his earlier this week after he returned from a vacation in Europe.
Even so, officials from the National League West champions say it's highly unlikely that they would offer more than a three-year contract.
Meanwhile, the Mariners remain as inactive in Nashville as practically every other team.
The meetings end on Thursday with the Rule 5 Draft. The Mariners have two openings on their 40-man roster.
"We have one name we're considering," Bavasi said, "but we're not sure which way we are going to go."
The Mariners selected right-handed reliever Sean White from the Pirates last year. A player selected in the Major League portion of the Rule 5 Draft must be on the 25-man roster throughout the season or be offered back to his original organization.