Inter confirmed the transfer in an official statement, explaining that the move occurred following the activation of the player's release clause.
Real Madrid have been determined to strengthen their defence this summer, and reinforcements at right-back became a priority after Dani Carvajal's departure from the club.
The Spaniard is now a free agent, with Onda Cero reporting that Inter Milan themselves could be an option for him following Dumfries's exit.
Dumfries, full name Denzel Justus Morris Dumfries, joined Inter from PSV Eindhoven in 2021 and went on to enjoy considerable success in Italy, including this season's Serie A title.
He made 28 appearances in all competitions during the campaign, totalling 1,176 minutes, and contributed five goals and two assists.
His release clause, set at €20million, made him an affordable option for interested clubs, and Barcelona were at one stage thought to be closing in on what would have been a bargain deal.
Liverpool and Manchester City also monitored his situation closely without success, before Real Madrid, a suitor few had anticipated, ultimately won the race for his signature.
Dumfries will compete for the right-back berth with Trent Alexander-Arnold, who joined the Bernabéu club only a year ago and has yet to fully convince, giving Real a genuine selection dilemma in that position.
He leaves Inter as a firm fan favourite, remembered for his part in some of the club's biggest nights over the past five years.
Those include his first goal for the club, a diving header, and a stoppage-time strike away at Lecce that sent supporters into raptures.
His only red card in that time came during the Milan derby, on the night Inter secured their latest Serie A star on the club badge.
He also scored twice in a Supercoppa Italiana win over Atalanta in Saudi Arabia, and produced a remarkable display in the Champions League semi-final against Barcelona, scoring twice, including an overhead kick, and setting up a goal for Marcus Thuram.
He was also seen greeting travelling Inter supporters after a difficult night in Munich, and was among those celebrating on the team bus during the club's Scudetto celebrations.
Dumfries now moves to LaLiga having previously played in the Eredivisie and Serie A, joining a Real Madrid side keen to strengthen their right flank as they look to return to the summit of European football.