The 24-year-old hasn't enjoyed the best of times since moving to Serie A in 2011, but Los Millonarios are apparently ready to take him back to Argentina
The 24-year-old attacking midfielder has struggled in Milan after joining from Velez Sarsfield in the summer of 2011, and has featured just eight times in Serie A this season.
Tunes admitted that there has been contact with the Nerazzurri over a possible switch, but no agreement has been reached as of yet.
"Alvarez interests us, and [club president Daniel] Passarella has already spoken with Inter officials," the vice-president was quoted in Tuttosport.
"[Coach] Ramon Diaz likes him, we'll see if Inter will give him on loan to us."
Nevertheless, the father of the Inter midfielder insisted that his son will either remain at Inter or seek playing time elsewhere in Europe.
"We are proud that a club as important as the River are interested, however, there is nothing official," Carlos Alvarez said.
"Ricky is finding his rhythm and he already knows that in February, he will be included in the Uefa list [for the knockout stages of the Europa League]. Of course, if Inter change their mind, it will do if loan offers come from Europe."