The Tottenham winger, who was named player of the year by the Professional Footballers' Association and the Football Writers' Association last season, is reportedly keen to speak to Real over a move to the Bernabeu.
Spurs have insisted Bale, who signed a new four-year contract at White Hart Lane in June 2012, is not for sale - and have said talks over a new deal are ongoing.
However, a report in Tuesday's edition of Spanish newspaper Marca claimed Tottenham are holding out for a world-record fee of 126million euros for the 24-year-old Welshman.
Having worked with Bale at international level, Wales coach Hartson has no doubt he could cope with a switch to the nine-time European champions.
"It wouldn't be an issue for Gareth," he said. "He's very level headed, his feet are firmly on the ground.
"I get to see him close up, obviously, working with him with the Welsh national team.
"He comes along to the squad and he works like all the other players, he gets his head down.
"He's got a great family. He doesn't come with a big entourage.
"He's got his agent, he's got his advisers and he's a really down-to-earth, level-headed lad and I think whatever Gareth does I'm sure he would have thought long and hard about it.''