The young defender has impressed since making his debut in April and has now caught the attention of the Champions League holders
The Catalans are scouting a number of young players with an eye on the future and have now added the 19-year-old to their shortlist, as they see the left-back as a potential replacement for Eric Abidal in the long term.
Boilesen is a product of the Brondby youth academy and moved to Ajax in January 2010. He made his official debut for the Amsterdam giants in the 3-0 win over Heracles in April 2011 and has so far made six Eredivisie appearances for Ajax.
The starlet, who is currently starring at the European Under-21 Championship in his native country, has a contract until the summer of 2013, but Ajax are expected to offer him a new and improved deal in the near future to ward off interest from the Primera Division giants.