The centre-back is a free agent having run down his contract with Portuguese outfit Porto this summer and there is understood to be significant Premier League interest in the 25-year-old.
According to the aforementioned source, West Ham and Newcastle have already made offers for the 53-cap star, along with Turkish outfit Fenerbahce.
Reyes is highly rated in his homeland, but missed the World Cup due to a hamstring injury.
He made the move to Portugal from América back in 2013 and went on to play 49 times for Porto.
In recent years he has been sent out on loan to secure more regular action, being a mainstay for both Real Sociedad in 2016/17 and Espanyol in 2017/18.