Oswalt, a free-agent right-hander, threw bullpen sessions for the Philadelphia Phillies last week and the Boston Red Sox and Texas Rangers this week, according to major-league sources.
He is scheduled to throw for one additional team next week, one source said. The identity of that team is not known.
Oswalt intends to sign with a team soon, possibly within the next week, and could pitch in the majors by mid- to late June, sources said.
The New York Yankees and Detroit Tigers are not in the mix for Oswalt at this time, sources said.
Oswalt rejected offers from the Red Sox and Tigers last offseason. His preference then was to play for either the Rangers or St. Louis Cardinals, but he has since decided to consider other teams.
The difficulty for the Rangers in signing Oswalt is the same as it was in the offseason; they lack an opening in their starting rotation. They have a sixth potential starter, right-hander Scott Feldman, in the bullpen. And the re-emergence of right-hander Koji Uehara (16 strikeouts, one walk in 14 1/3 innings) conceivably could give them the flexibility to return righty Alexei Ogando to the rotation, if needed.
Another possibility for the Rangers would be to trade right-handed starter Colby Lewis, who is 3-3 with a 3.60 ERA but is tied for the major-league lead with 12 home runs allowed. However, there is no indication that the Rangers are considering such a move, and there is no guarantee that Oswalt would pitch better than Lewis.
One source said that is Oswalt is in good shape, but not surprisingly, more in spring-training mode than mid-season form.
Oswalt, 34, went 9-10 with a 3.69 ERA in 23 starts for the Phillies last season, missing six weeks with lower back inflammation. The Phillies declined his $16 million option at the end of the season.