The Denmark international has 15 caps for his country, and is eager to leave the Eredivisie for a new challenge with just one year remaining on his contract.
The Dutch giants are said to have rejected Villa's first offer, said to be around £1.3 million, but have been told by Ajax that another £1 million added to that will be enough to do business.
Boilesen, who spent the last few months as captain of the club, first broke into the senior team in 2011, but only made limited appearances in his first three seasons, with an injury hit 2012-13 season hindering his development somewhat.
But the Dane has been much more involved in the starting eleven in the last couple of campaigns, making 28 appearances or more in the two seasons.
Despite his importance at the club, the Mail have quoted Ajax director Marc Overmars as saying Boilesen is eager to move to a foreign league, with one year remaining on his contract: ”He has indicated that he wants to look abroad", said the formal Arsenal star. "And that is fine.”
Ajax have decided to bring Mitchell Dijks back to Amsterdam after one year away, making it very likely that the Villa target will be allowed to move this summer if a suitable fee is reached.
What does this mean for Villa's fullback options?
There are already four left back options in the first team squad at Villa Park, but all three have some kind of questions marks over them.
Kieran Richardson, much more happy as a winger, is 31 this year, and missed sections of last season with injury, while Aly Cissoko fell out of favour once Tim Sherwood took charge, making just one appearance under the ex Spurs manager.
Both Joe Bennett and Antonio Luna were out on loan last season, and neither are expected to be part of Sherwood's squad next season.
Which may well explain Villa's chasing of a player who has won four league titles during his time with Ajax.