Eriksen, who last month reportedly turned down an approach from newly-promoted Championship club Wrexham, has otherwise expressed that he is mostly keen on continuing his career away from the English Premier League.
The Dane left Manchester United at the end of last season, where he struggled to maintain a regular spot in Ruben Amorim's starting eleven.
Last month, Eriksen’s representative Martin Schoots told Mail Sport: "Christian still wants to play in a first-tier competition and he absolutely wants to stay in Western Europe. Some contacts are going on, but it has to suit both sides."
Back in March, Christian Eriksen spoke about his dreams for the future where he rejected a move to destinations like Denmark and the United States, but he was generally very open about where he would continue his career.
However, he also expressed his desire to play at a high level so that his dream of participating in the 2026 World Cup could be fulfilled.
Burnley have spent a little more than £60million on eight new players this summer. Just like Eriksen, current Burnley boss Scott Parker played for Tottenham, but left in the summer of 2013 before Eriksen arrived from Ajax