The English club is building a project aimed at an immediate return to the top flight and has identified Lopez as one of the profiles under consideration.
Sky Sports UK reports that Burnley have approached Genoa about Lopez, although the sporting director has not yet given a definitive answer.
Lopez is understood to be comfortable at Genoa and believes in the project being developed by the club. Daniele De Rossi is in charge of the team, while the wider management structure is working on plans for the future.
The interest from England therefore presents Lopez with an opportunity, but there is currently no decision on whether he would leave Italy.
Lopez has also informed Genoa about the approach from Burnley. That step reflects the relationship he has established with the Rossoblù and leaves the situation open while both sides consider what happens next.
For Burnley, the attraction is clear within the context of their plans. The club wants to return to the Premier League at the first opportunity after last season's relegation and is assessing candidates who could help shape its sporting strategy.
Lopez now has to weigh that opportunity against his current position at Genoa, where he remains focused on the project. The fact that he has not rejected the approach means the possibility of a move remains open, but there is no agreement at this stage.
The next phase will depend on whether Burnley decide to advance their interest and whether Lopez is prepared to consider a change. For now, the Genoa sporting director has made his club aware of the approach and continues to work on the Rossoblù project.
The situation could develop in the coming days, with Burnley needing to establish whether there is scope for further talks. Lopez, meanwhile, will have to decide whether the prospect of helping rebuild an English club represents the right next step in his career.