The news could come as a disappointment to Tottenham and Juventus, who are both reportedly interested in signing the Uruguayan striker, although it is not clear whether Real would be interested in the 31-year-old.
The daily Marca said the offer did not come to Real Madrid directly but through an authorized intermediary for Atletico.
The proposed price tag of 18 million euros (25 million dollars) would be just half of his buyout clause, and is negotiable, it said.
"Atletico intends to sound out the possibility of Forlan's transfer in January for one reason above all: the request of its coach Quique Sanchez Flores," said Marca, which did not quote any sources in the report.
Forlan has had a disappointing start to this season, scoring just three league goals, the last on September 11.
Marca said Real "for the moment is not considering buying Diego Forlan," but at the club "they have learnt to never say never."
Real coach Jose Mourinho said last month that he had no plans to sign more players during the winter transfer window.Forlan left Manchester United - where he had two inconsistent seasons - in 2004 for Villarreal, who then sold him to Atletico in 2007 for 21 million euros.
He has since been an integral part of the Atletico side, which won the UEFA Europa Cup last season with the Uruguayan scoring both goals in the club's 2-1 defeat of English side Fulham in the final.
The player has twice won the award as the Spanish Primera Liga's top scorer and twice won the European Golden Boot.
He was also awarded the Golden Ball as the best player of the 2010 World Cup after grabbing five goals in tournament, in which Uruguay finished in fourth place.
Forlan indicated in an interview with Marca last month that he could leave Spain for the English Premiership if the right offer comes along.
