Reus, who is currently sidelined with the ankle injury that kept him out of Germany's successful World Cup campaign in Brazil, is contracted to Dortmund until 2017.
However, this deal is also understood to include a release clause of around £27million, which can be activated from next summer.
As a result, Dortmund have tabled a new contract offer with an improved wage, but cancelling out the initial release clause, reports in the German media claim.
The country's biggest national newspaper, Bild, understand that Reus has rejected the contract offer, giving the likes of Liverpool and Barcelona, two of the clubs supposedly chasing the 25-year-old forward, fresh impetus in their hopes to sign him.
Following the reports in Germany, Sky Bet have renewed the odds on Reus' next club, with Liverpool heading the billing.
The Merseysiders have been cut as short as 15/8 favourites, with Bayern Munich next at 4/1. Manchester United (7/1), Arsenal and Barcelona (both 14/1) are also possible destinations for the former Borussia Monchengladbach star.
Brendan Rodgers has brought in six new faces at Liverpool this summer - half of those from Southampton - in an attempt to replace their top goalscorer of the last three seasons, Luis Suarez, who joined Barcelona for £75million.
A deal for QPR striker Loic Remy collapsed over reported issues with the Frenchman's medical, while a deal for Belgium forward Divock Origi looks to be close.