Hayes, 29, has scored six goals in 22 matches for the Scottish Premiership side this season, including one in the Europa League.
He won the 2010 Scottish First Division title with Inverness and then the 2014 Scottish League Cup with Aberdeen.
Cardiff have already signed utility player Greg Halford from Championship rivals Rotherham and striker Ibrahim Meite from non-league Harrow Borough.
Now they are trying to add Hayes, who has two caps for the Republic, as manager Neil Warnock looks to revamp his squad in the January transfer window.

Elliott Holdham, who has Asperger's syndrome, heard about the competition on social media and was desperate to be involved.
His dad called up the local organisers, who invited them to watch AFC Dunstable in round one of the tournament.
On the day, the team were a player short and the 19-year-old stepped in, scoring six goals and securing a place in the side for the second round.
I am hoping to push onto bigger and better things
Holdham's father, Jason, said: "I'm really proud. Football has helped Elliott grow in confidence and he's been a breath of fresh air for the team."
His son has been a Luton Town supporter since he was six but only began playing football eight months ago.
Due to his condition, Holdham had not found a suitable team to play for, until discovering the FA People's Cup.
AFC Dunstable's manager Josh Spavin said they were "thrilled to find such a great player".
Speaking on BBC Three Counties sports programme with Geoff Doyle and Luke Ashmead, Spavin added that "without Elliott they would have struggled over the weekend".
And, even though AFC Dunstable did not manage to get through to the finals day, Holdham has high hopes for the future.
"I am hoping to push onto bigger and better things," he said.
You can catch up with the highlights of the FA People's Cup semi-finals here and will be able to follow the finals day on Sunday, 20 March with our live text commentary.

The 'scrambler' fell from Bristley Ridge, between Tryfan and Glyder Fach at about 15:00 GMT on Saturday.
Two climbers tried to rescue the man but could only secure him to the rock face, Ogwen Valley Mountain Rescue told BBC Wales.
He was airlifted to Ysbyty Gwynedd where he was pronounced dead.

Goodwin, who gave Trevor Francis his Blues debut, led the club to two FA Cup semi-finals and promotion to the old Division One in his time in charge, from June 1970 to September 1975.
He had previously been manager of both Scunthorpe United and Brighton.
During his playing career, he was a wing-half for Matt Busby's United, then Leeds before finishing at Scunthorpe.
As one of the Busby Babes, he made his first team breakthrough at Old Trafford following the Munich Air Disaster in 1958.
He played in the first game after Munich, just a fortnight later, when the team-sheet in the programme was famously left blank for the club's FA Cup fifth-round win over Sheffield Wednesday.
And he was in the rebuilt United team all the way to Wembley, where they lost 2-0 to local rivals Bolton Wanderers.
Heywood-born Goodwin also played first-class cricket over two summers, making 11 appearances as a fast bowler for his native Lancashire, in which he took a career-best 5-35 against Middlesex at Lord's in June 1955.
After leaving Birmingham in 1975, Goodwin took up an offer to move to the then burgeoning North American Soccer League, where he had two spells as coach, then club president of the Minnesota Kicks.
He remained in the United States for the rest of his life, and died at home in Gig Harbor, Washington State at the weekend.
Birmingham City will pay tribute at Tuesday night's home game with Bolton Wanderers at St Andrew's. The players will wear black armbands and there will be a minute's applause.
Cardiff City are targeting Aberdeen's Republic of Ireland winger Jonny Hayes.
[NEXT_CONCEPT]
The FA People's Cup regional semi-finals saw 400 teams compete across eight venues on 27-28 February - and for one player it was particularly special.
[NEXT_CONCEPT]
A man has died after falling a "considerable distance" from a Snowdonia mountain.
[NEXT_CONCEPT]
Former Birmingham City manager and Manchester United player Freddie Goodwin has died at the age of 82.