Hearts value the Scotland Under-21 international at around £1m.
The 23-year-old attacker is contracted at Tynecastle until next summer and has turned down the offer of an extension.
Walker, who has made more than 180 appearances for Hearts and who scored 15 goals last season, can sign a pre-contract agreement with another club from 1 December.
Rangers' initial bid was rejected on Wednesday.
Hearts director of football Craig Levein said last month that Walker wanted to leave the club.
The club subsequently stated "the player would prefer to see out the remaining year of his contract".
Rangers finished third in the Scottish Premiership - two places and 21 points above Hearts - and have signed defenders Bruno Alves and Fabio Cardoso, midfielder Ryan Jack and forwards Daniel Candeias and Dalcio this summer.
The club, who will enter the first qualifying round of the Europa League later this month, have also had a second offer for Norwich City midfielder Graham Dorrans turned down.
Hearts have rejected a second bid from Rangers, believed to be in the region of £600,000, for winger Jamie Walker.