The 29-year-old is due to travel to England for a medical on Monday after the Bundesliga club accepted an offer worth an initial £6.8m, with a further £1.7m in potential add-ons, according to the BBC.
Elvedi's move ends an 11-year association with Borussia Monchengladbach, where he progressed through more than a decade of first-team football after joining from FC Zurich, and reunites him with Leeds United manager Daniel Farke, who previously worked with the defender in Germany.
The versatile centre-back has earned 73 caps for Switzerland and was part of the squad that competed at this summer's World Cup, giving Farke's promoted side a considerable upgrade in experience at the back ahead of their second successive Premier League campaign.
Asked about the deal after Leeds's 4-0 pre-season win over Augsburg, Farke was cautious about confirming the transfer publicly.
"I've been working in football long enough to know everything is only final once the medical checks are completed and all the paperwork is signed. We've seen situations like this before," he said.
"So I only speak about new players once they have been officially announced and everything is settled. But I don't want to lie, and we expect something to happen in the next 48 hours."
Elvedi's departure is also set to trigger a further move for Leeds, with fringe defender Sebastiaan Bornauw poised to join Hamburg on an initial loan that includes an option to buy, freeing up space in Farke's squad following the Swiss international's arrival.
The defender was left out of Gladbach's squad for a recent friendly against Aston Villa as the move edged closer, with Sky Germany reporting the German club will also receive a 15 per cent sell-on clause as part of the agreement. Gladbach head coach Eugen Polanski confirmed before that game that Elvedi had been given permission to hold talks with a club, having previously named the defender vice-captain in an attempt to persuade him to sign a new contract.
Elvedi made 364 appearances in all competitions during his time at Gladbach after joining from FC Zurich in 2015, establishing himself as one of the Bundesliga club's most consistent performers across more than a decade in Germany.
Leeds begin their Premier League season away at Nottingham Forest next Saturday, with Farke expected to have his new centre-back available for selection once the move is formally completed in the coming days.