England won the match in Podgorica 5-1 but the match was overshadowed by racist chanting from some home fans directed at several England players, including Rose.
Uefa said "disciplinary proceedings" had been opened against Montenegro with one charge for "racist behaviour".
The case will be dealt with by European football's governing body on 16 May.
Montenegro coach Ljubisa Tumbakovic said he did not "hear or notice any" racist abuse.
But England manager Gareth Southgate, speaking to BBC Radio 5 Live said he "definitely heard the racist abuse of Rose".
"There's no doubt in my mind it happened," he added. "I know what I heard. It's unacceptable.
"We have to make sure our players feel supported, they know the dressing room is there and we as a group of staff are there for them.
"We have to report it through the correct channels. It is clear that so many people have heard it and we have to continue to make strides in our country and trust the authorities to take the right action."
Montenegro also face other charges relating to crowd disturbances, the throwing of objects, setting off of fireworks and the blocking of stairways following the game at the Podgorica City Stadium.
BBC Radio 5 Live commentator Ian Dennis said he heard racist chants when Tottenham left-back Rose was in possession of the ball. Raheem Sterling and Callum Hudson-Odoi were also subjected to abuse.
In injury time Rose was booked following a strong challenge on Aleksandar Boljevic, with more racist chants aimed at the 28-year-old.
It is the latest incident in a series of racist incidents involving Rose and his family.
In 2012, following an under-21 game against Serbia the country’s FA was fined £65,000, with their under-21s having to play a game behind closed doors after he was subjected to racist abuse during a game.
Last summer, he revealed he had advised his family against travelling to the World Cup in Russia over fears that they would be subject to racism and also revealed that his mum had been subjected to racist abuse in Doncaster, which had caused him to suffer from depression.
Chelsea winger Hudson-Odoi, 18, told BeIn Sports: "When you are hearing stuff like that from the fans, it is not right and it is unacceptable. Hopefully Uefa deal with it properly. When me and Rosey went over there, they were saying, 'ooh aa aa' monkey stuff and we just have to keep our heads and keep a strong mentality.
"Hopefully Rosey is OK too. We will discuss it and have a chat. He has a strong mentality and is a strong guy so hopefully everything will be good.”
Born in Doncaster in 1990, Rose started his professional career at Leeds United in 2006 and played for Watford, Peterborough United, Bristol City and Sunderland before establishing himself at Tottenham and earned his first senior England cap in a 3–2 victory against Germany in 2016.
Last year, he revealed he had been diagnosed with depression after his uncle killed himself and his mum Angela was subjected to a racist attack.
He said: “She was very angry and upset about it, and then someone came to the house and nearly shot my brother in the face. It was a testing time.”