The 30-year-old has largely found it difficult to make his mark for Leeds since arriving in August 2020, but the attacker enjoyed a successful six-year spell in La Liga with Valencia.
According to The Athletic, Barcelona contacted Leeds towards the end of the transfer window to determine whether it would be possible to sign Rodrigo on loan.
However, the report claims that the Whites made it clear that the Spain international was not available on a short-term basis.
Rodrigo, who still has three years left to run on his contract at Elland Road, has scored seven times and registered two assists in 32 appearances for Leeds in all competitions.
Barcelona allowed Antoine Griezmann to re-join Atletico Madrid on loan before the window closed and moved to sign Sevilla striker Luuk de Jong as a replacement.