SEATTLE — The effort by a California billionaire named Thomas F. Steyer to bolster global climate change measures in Washington has turned the battle over the State Senate into one of the most expensive legislative elections in state history.

Money has poured into the handful of legislative races that Mr. Steyer’s political action committee identified as central to shifting the Senate’s leadership from a Republican-led coalition to a Democratic majority that would support the ambitious climate goals set by Gov. Jay Inslee, a Democrat.

About $4.2 million has been spent so far by independent groups on the 10 most competitive state legislative races, up from $3.2 million for the top 10 in 2012 and $1.8 million in 2010, according to state figures. In terms of spending by outside independent groups, this is almost certainly the most expensive legislative election season in state history, the records show.

The Democrats need a net gain of two seats to achieve a Senate majority, and Mr. Steyer’s political action committee, Nextgen Climate Action, has contributed $1.25 million to that goal.