Kapino has become a free agent after being released from his deal at OIympiakos in December.
The 23-year-old has nine international caps for Greece and is reportedly in advanced talks with the club.
Greek outlet Sport.fm claimed Kapino is already in England and is in advanced talks with the club.
Kapino would be the second goalkeeper to join Forest in the past month, following loanee Costel Pantilimon. It would be an ominous sign for Jordan Smith.
Forest do not necessarily need Kapino, but bringing him on board might be too good an opportunity to ignore and manager Aitor Karanka is clearly keen to shake things up and add competition across the squad.
Karanka has inherited a squad which narrowly avoided relegation last season and have languished in the bottom half of the table this campaign, with Mark Warburton failing to take them forward.
Kapino will become another useful option for the former Middlesbrough boss in his efforts to build some momentum ahead of an attempt to make a real push for promotion next season.
Here is a look at how Forest fans reacted on Twitter to the reports...
Kapino could be a good singing still only 23 had big clubs chasing him a few years ago because of his potential but I reckon it could be a good signing for us so he hopefully get his career running again and succeed his potential #Nffc— ً (@Kieranffc18) February 7, 2018
Kapino would be Competition for Smith next season. 3 way battle can only help developments. #nffc— Cameron Clark (@Camclarky) February 7, 2018
Kapino would be a quality signing on a free! Had big clubs looking at him in the past! Would definitely be first choice at #nffc also would be good to send Smith out on loan— Michael Cleaver™ (@NFFCCleaver) February 7, 2018
Kapino would walk into a premier league team. Had big clubs sniffing around a few years ago. Be quite a coup on a free. Top quality keeper. #nffc— Mr Chef (@ShaunBrianBooth) February 7, 2018