Both parties are keen for the club captain to extend his deal at the Santiago Bernabeu with Ramos' contract expiring in June 2021. The defender would be free to negotiate with other clubs from January.
In June this year, ESPN reported that both sides had agreed to delay negotiations until the end of the summer as the club assessed the financial effects of the coronavirus pandemic.
When the extent of the impact became clear, Real opted not to spend in the summer transfer market and put talks with Ramos on hold too.
However, sources have told ESPN that in recent weeks the club and Ramos' representatives have started to sound each other out over discussions on a contract renewal which is viewed as a priority.
The feeling from those initial talks is positive, and there is a mutual desire for the 34-year-old to extend his stay at the Bernabeu.
Sources told ESPN that Ramos would like to stay at Madrid until June 2024.
With his form and fitness record, the club would not be against that, although it remains to be seen if they will agree year-on-year renewals, or find a longer-term agreement.
Real tend to offer year-long contract extensions for players over the age of 30, but there is an acceptance that Ramos is "special" given his importance on and off the pitch.
"He is our captain and our leader, of course we want him here forever," coach Zinedine Zidane said on Tuesday, after Ramos scored in a 3-2 Champions League win over Inter Milan. "I have no doubts he will stay and keep making history."
President Florentino Perez said Ramos "will be here for life" after Madrid won La Liga in July.
Ramos arrived at Real Madrid from Sevilla aged 19 in 2005 and is now in his 16th season at the Bernabeu. He has won La Liga five times and the Champions League four times at the club as well as the World Cup and two European Championships with Spain.
He finished last season with 11 league goals, the club's second top goalscorer, and his header against Inter was his 100th strike in a Real Madrid shirt.