The Rangers insisted they had made the one-year offer Messier wanted in the first place.
Rice's only other option at the time was to accept a one-year, $7.5 million offer from the Chicago Bulls.
They are believed to have raised their one-year offer to about $2.5 million from $2 million.
The suit estimates the tax-adjusted value of the one-year offer worth $650,000 to $700,000.
Ventura would be less likely to accept a one-year offer from another team.
Shinjo accepted a one-year offer, with two option years, that guarantees him only $700,000.
Last week, the team improved its one-year offer to the former high school pitcher.
When the possibility of a one-year offer was presented to Martinez, he was not thrilled.
Foster, 26, an unrestricted free agent, turned down a reported one-year, $1 million offer from the Bengals three weeks ago.
Last week he accepted the Marlins' one-year offer of $10 million.