Tamudo, who previously played for the club between 2011 and 2012, said that he has "all the desire in the world" to impress for Rayo.
"I am here with all the desire in the world," Sky Sports News quotes him as saying. "Now I hope to be another player who can help, like I always want to do. The team has been doing well and now I hope that I can help them go on to celebrate something.
"This is a moment to enjoy now the team are doing well and pushing for the positions at the top end of the table.
"When [sporting director] Felipe Minambres called me I was pleased because it was unexpected and I had thought that there wouldn't be any more chances for me to play in Spanish football."
Ray are currently seventh in La Liga.