According to Mundo Deportivo (h/t Naveen Ullal of Sport Witness), Spurs are keen to tie down the Danish playmaker to a new deal, with his contract set to expire in 2020. It's added the offer would be as "important" as the one star goalscorer Harry Kane signed in the summer.
Mundo Deportivo also reported Eriksen has yet to agree to an extension, which is a result of interest from the European champions. It's added there's a belief the longer the Denmark international stalls, the more likely it is that he could leave.
The former Ajax man's agent, Martin Schoots, is said to be aware that Madrid are considering moving for the player in the future.
Cadena COPE (h/t Ullal) reported that Eriksen is being looked at as a possible long-term replacement for Luka Modric, who also left Spurs for the Spanish capital. The Croatia international has established himself as one of the best midfielders in the world during his time in Madrid and was recently crowned FIFA's Best Men's Player.
Eriksen is not at that level, although there's no doubt he has improved under manager Mauricio Pochettino at Tottenham.
Not only is he a creative hub for Spurs, but he is also adept at finishing chances:
Tottenham have a number of excellent attacking players, with Kane, Dele Alli, Lucas Moura and Heung-Min Son all able to decide games with their class. However, when Eriksen is out of the team, they struggle for rhythm.
Typically tucked in from the right flank, the playmaker is often the man Tottenham look to in the final third.
While Spurs have enjoyed a strong start to the Premier League season and sit just two points off the top, James Gheerbrant of The Times commented on how one-dimensional the team can appear without Eriksen in the starting XI:
James Gheerbrant @JamesGheerbrant
The absence of Eriksen and Alli really exposes the almost total lack of creativity and penetration in Spurs' passing from central midfield. Do we have to add 'noticing Mousa Dembele was past it before anyone else did' to Roberto Martinez's list of achievements as Belgium manager?
The 26-year-old hasn't been in action for Spurs since September 22, when the team beat Brighton & Hove Albion 2-1.
There were some fears the injury he had was serious after comments from Denmark manager Age Hareide. However, as Kieran Gill of MailOnline relayed, Eriksen's agent said the player has "no chronic injuries" and would be "back as soon as possible."
While Modric is approaching the twilight of his career, he has traditionally operated in a more withdrawn role than Eriksen. If the Dane were to join Madrid at the end of the season, he would likely be competing with Isco, Marco Asensio and Lucas Vazquez for a spot in the side.