The 28-year-old is set for a switch to the Premier League in search of regular first-team minutes, having failed to feature for Diego Simeone's men in a competitive match during the 2023-24 term so far.
Grbic has played second fiddle to first-choice keeper Jan Oblak, however the Croatian has experience of Ligue 1 and Champions League football from his time at Lille in the 2021-22 campaign.
Since securing a switch to the Estadio Civitas Metropolitano from NK Lokomotiva in his homeland during the summer of 2020, the former Hajduk Split man has appeared just 14 times for the Spanish giants.
According to the Daily Mail, Sheffield United are closing in on their second signing of the January transfer window in the form of Grbic.
The report claims that the shot-stopper flew into England on Tuesday in order to finalise a switch to the Blades, with a medical booked in for this week.
Chris Wilder's side will supposedly pay Atletico £2m for their two-time international, whose contract in the Spanish capital expires at the conclusion of the campaign.
Having conceded 51 goals in their opening 21 Premier League matches, it is no secret that the Blades have been looking to bolster their goalkeeping and defensive ranks this month.
Rotherham United's Viktor Johansson was linked with a move to Bramall Lane earlier in the window, as well as former Leicester City stopper Kasper Schmeichel.
However, it is understood that United have settled on the additional of Grbic, and the 28-year-old will join Ben Brereton Diaz as a winter arrival in Yorkshire.
Wilder has expressed his desire to immediately improve his starting XI with incomings during this transfer window, meaning Wes Foderingham's place in between the sticks is set to be under threat.
The 33-year-old has managed just one clean sheet in 21 top-flight starts this season and will see his current terms with the Blades expire in the summer, although it has been reported that the keeper is in discussions with the club over an extension.
Subject to Grbic passing his medical and receiving international clearance, the Croatian could make his Blades debut in the FA Cup against Brighton & Hove Albion on Saturday afternoon.