Despite having two years left on his contract with the Super League side, the 23-year-old made the switch from league to union and joined Sale on a three-year deal.
Castleford are now seeking to claim damages against Sale, Solomona and Andy Clark, his agent, at the High Court in Leeds.
"The proceedings have been issued as a last resort," Tigers chairman Steve Gill said. "We have failed over the last couple of weeks to try to resolve matters without the formality of court proceedings.
"We have confidence that the legal system will bring out all of the facts and the truth. We are fighting this not just for Castleford Tigers but also for the integrity of all sports, including, of course, rugby union clubs.
"Our belief is that all contracts are sacrosanct in professional and amateur sports."
Solomona, a Samoa international, scored a record 42 tries in Super League last season.