The 25-year-old forward has been linked with a move to the Premier League, with Chelsea thought to be interested, along with Newcastle and Swansea.
But the Berkshire club have reportedly moved first, and according to a Daily Mirror source, are the only club to have made an official offer for the Spaniard so far this month.
Aspas is thought to have a €10m release clause in his Celta Vigo contract, but Reading have not meet that fee.
The Daily mirror source said: "At the moment there is no agreement with any club. There has only been one official offer, from Reading…but no offer has arrived from any team that meets the clause."
The La Liga club are keen to help hold of Aspas as they look to stave off relegation to the Spanish second tier.
But the clause in his contract leaves Celta powerless to stop him leaving if a club matches the €10m fee.
Aspas has won plaudits for his goalscoring - he has eight goals in the league so far - and his ability to create chances for others in a struggling side.
Celta sit 15th in La Liga, just two points clear of the relegation zone, and the Spanish club are desperate for the transfer window to close before any club triggers Aspas's release clause. But Reading are looking for striker reinforcements as they face a relegation battle of their own in the Premier League.