Roy Oswalt has thrown for the Phillies, Rangers, Red Sox and at least one other team this month.
However, the free-agent remains unsigned as June approaches.
Major League executives claim that teams have been turned off by the right-hander's asking price.
The Dodgers checked in and considered the price "excessive."
One source said Oswalt wants a pro-rated $7.5 million in base salary. Another said that Oswalt's desired salary is a moving target, "but always high."
Teams Turned Off By Oswalt's Asking Price