Liverpool have their sights firmly locked on the Estadio Santiago Bernabeu currently, with two Real Madrid stars interesting the Premier League club.
Interest in Asensio has been known for some time, but now the Merseyside outfit are ready to take on several other suitors to try and land Vazquez.
New Arsenal head coach Unai Emery is understood to be a big fan of the 27-year-old, while Chelsea are also keeping a close eye on developments.
AC Milan were also thought to be in the hunt, but what effect their two-year ban from European competitions will have on that remains to be seen.
Vazquez and Asensio operate in reasonably similar positions and under Jurgen Klopp would both be likely to be fielded in one of the wide roles in his three-man attack.