The Blaugrana had hoped to sign the Spain striker on a six-month loan with no obligation to make the deal permanent in the summer, but the Mestalla outfit were holding out for a fee of around €60 million.
Portuguese super agent Jorge Mendes, who is close to Valencia owner Peter Lim, was in Barcelona for the talks and tried to unblock the deal by setting up a transfer for Sporting CP midfielder Bruno Fernandes to the Catalan club this month.
As part of the arrangement, the 25-year-old was then to spend the rest of the season at Mestalla.
That offer tempted Valencia and meant Rodrigo would have been free to move to the Camp Nou on loan, but the Portuguese midfielder is now close to joining Manchester United.
With Valencia unwilling to let their striker leave for only a loan fee in the middle of the season and Barca not prepared to pay €60m for a player who may have only been a short-term replacement for the injured Luis Suarez, the Catalan club ended talks on Tuesday evening.
According to reports in the Catalan media, Barca could now turn their attentions to signing Ajax striker Dusan Tadic on a six-month loan deal.