The 22-year-old has been on World Cup duty with Serbia, but the Orlovi were dumped out on Wednesday by Brazil, meaning there could now be some acceleration on the proposed deal.
Grujic has found opportunities difficult to come by at Liverpool since joining in 2016 from Red Star Belgrade for £5.1 million, and now his chances are even more limited with the summer signings of Fabinho and Naby Keita.
To date, Grujic has only ever made eight substitute appearances in the Premier League, and BBC Sport understand Liverpool would be happy to allow the youngster to leave in order to get some regular game time.
Cardiff will be hoping to get the deal over the line as a follow up to the recent signings of Bobby Reid, Alex Smithies, Greg Cunningham and Josh Murphy.
It's been a promising start to the summer transfer window by Neil Warnock, but one gets the feeling that the Bluebirds will need a marquee name if they are to avoid relegation come May 2019.