Marca claimed that a number of European sides had travelled to watch Valverde in action for Penarol, but said the Gunners were the only team to have opened negotiations so far.
It was reported that the 17-year-old has even flown over to the UK to train with the north Londoners.
Real, Barcelona, Manchester United and Chelsea were also listed as suitors by Marca, who said that a decision over Valverde's future would be announced upon his 18th birthday in July.
However, in May last year it was reported by AS that the midfielder has already signed for Los Blancos, with his arrival in the Spanish capital delayed until he reaches the aforementioned landmark.
The report claimed that Real have paid a fee of five million euros for Valverde, who watched his future club in action during last season's Champions League semi-finals, after undergoing a medical.
AS described the youngster as 'the pearl' of Uruguay's Under-17 side.