The Reds decided to sign Arthur on loan from Juventus before the deadline, with the club suffering injuries to the likes of Thiago Alcantara, Jordan Henderson, Naby Keita, Fabio Carvalho, Curtis Jones and Alex Oxlade-Chamberlain.
According to AS, Jurgen Klopp's side made a bid in the region of £86.5m for Valverde, with the club determined to bring him to Anfield before the deadline.
However, the report claims that Los Blancos found the offer to be 'offensive' and made it clear that they would not be selling him under any circumstances.
Valverde joined Real Madrid from Penarol in 2016, initially representing Real Madrid Castilla before making his first-team debut in 2017.
The Uruguay international has represented the Spanish outfit on 155 occasions in all competitions, contributing eight goals and 10 assists in the process.
Valverde was a vital player for Carlo Ancelotti's side during the 2022-23 campaign, making 46 appearances in all competitions, contributing one goal and two assists in the process.
The South American started the campaign as a central midfielder but was used as a right-sided attacker in the latter stages of the season, starring in that position in the Champions League, with Los Blancos winning the competition for the 14th time in their history.
Valverde has also been key for Real Madrid this term, making seven appearances, contributing two goals an two assists in the process.
The Uruguayan dropped back into central midfield for Sunday's La Liga clash with Mallorca and found the back of the net with an impressive goal in the 45th minute of the contest.
Valverde has a contract at Bernabeu until June 2027, placing the reigning Spanish and European champions in a strong position when it comes to his future.
Klopp revealed late in the summer transfer window that Liverpool had been in the market for a midfielder who ultimately joined another club.
There have been suggestions that the German was talking about Aurelien Tchouameni, who moved to Real Madrid from Monaco, but the Reds head coach did not specifically mention the player in question.
"We were going for midfielder but that midfielder decided to go to another club. That can happen. Some players are not available now but for us really interesting. That's how it is - so we can make that decision," Klopp told reporters.
Liverpool have only won two games in all competitions this season and suffered a 4-1 defeat to Napoli in the Champions League last time out.
The Merseyside giants will not be back in Premier League action until October 1 against Brighton & Hove Albion due to the suspension of two league games, but they will face Ajax in the European Cup on Tuesday night.