The 59-year-old from Leeds, West Yorkshire, was arrested in Ms Cooper's constituency Castleford on February 1.
A West Yorkshire Police spokesman said he was held 'as a result of information received in relation to alleged threats against a serving MP'.
The force confirmed the man was bailed and that today (Tuesday) he accepted a conditional caution.
The spokesman refused to disclose the exact nature of the threat.
Ms Cooper, the MP for Normanton, Pontefract and Castleford, received a threat against the lives of her children and grandchildren in 2016, shortly after the murder of Jo Cox.
A tweet from a constituent angry at receiving Remain literature said: "Please stop or I will kill your kids and grandkids."
The former secretary of state contacted the police on that occasion