Eriksen, 29, is a free agent after his contract with Inter Milan was terminated last month following the Italian medical authority's refusal to allow him to play with a cardiac device implant.
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He was subsequently fitted with an implantable cardioverter defibrillator (ICD). Serie A rules prohibit any player using such a device, but Eriksen believes he can continue to play at the highest level.
And Brentford are believed to have offered a six-month contract with the option for a further year. Negotiations are said to be at an early stage and sources have told ESPN that a number of clubs in England and other countries are talking to Eriksen's representatives about a possible deal.
Brentford manager Thomas Frank previously coached the midfielder while working for Denmark with the under-17 team.
The midfielder began training with Odense Boldklub, a Danish side he spent three years with as a youth team player, before working with Swiss third-tier team FC Chiasso.
Sources have told ESPN that Eriksen is open to a return to England having spent seven years at Tottenham before joining Inter in January 2020.
Earlier this month, Spurs manager Antonio Conte said "the door is always open" for Eriksen to return to the north London club.
In an interview with Danish broadcaster DR1 earlier this month, Eriksen revealed his aim of playing for Denmark at the World Cup, which begins in November.
"My goal is to play in the World Cup in Qatar," Eriksen said. "I want to play. That's been my mindset all along. It's a goal, a dream. Whether I'll be picked is another thing. But it's my dream to come back. I'm sure I can because I don't feel any different. Physically, I'm back in top shape.
"My dream is to rejoin the national team and play at [Denmark's national stadium] Parken again and prove that it was a one-timer and that it won't happen again.
"I want to prove I've moved on and that I can play [for] the national team again. Again, it's up to the manager to assess my level. But my heart is not an obstacle."
Eriksen would have to be assessed by a Football Association-approved cardiac specialist before he could sign for an English club. Around 1,500 such examinations are done each year, with the FA providing a central storage system for screening results which FA medical staff and authorised clubs have access to.
An FA spokesperson said: "In England, any player that has an abnormal cardiac screen or who develops a cardiac problem would be assessed by a sports cardiologist. We would expect the sports cardiologist to be a member of The FA Cardiac Consensus Panel, a group of experienced sports cardiologists who advise The FA with regard to these issues and provide consultation and screening expertise for our cardiac screening programmes in professional football.
"This would also be done in association with the team doctor who usually looks after them. The cardiologist would look at the individual circumstances and the risk surrounding the player and they would make a decision on whether the player could continue to play or should stop.
"The FA would not ban someone from playing based on a cardiac screen because a) medical information of this kind is confidential between the player and the doctor who is caring for them and b) any risk is to the player themselves and not to others on the pitch, so it is an individual decision which the player takes with support and advice from those who are responsible for their personal medical wellbeing.
"In Italy, there is a longstanding nationwide cardiac screening policy which runs across all sports participation and does not allow participation in a case of increased risk of sudden cardiac arrest during exercise. This has not been adopted in other countries, including the UK.
"With regard to Christian Eriksen playing in England, it is impossible to comment on his individual circumstances without knowledge of his condition and the risks associated with it. As always, any assessment would be on an individual basis."