The centre-back impressed in the club's Ligue 1 campaign last season that the Sir Jim Ratcliffe-owned side qualify for the Europa League.
A move to Manchester United - another of Ratcliffe's assets - was expected to follow only for the transfer to be blocked by UEFA due to rules on multi-club ownership.
United's sudden exit from negotiations paved the way for other top European sides to open discussions with Nice for the defender who previously joined Barcelona as a youngster.
Juventus were keen to secure his services as Thiago Motta attempts to overhaul the Italian side's squad and Todibo was eager to make the move to Turin despite Premier League interest.
West Ham technical director Tim Steidten has travelled to France to try and conclude negotiations for Todibo
Their advancement in negotiations with the French side appeared to have ruled the Hammers out of making a move, but Kurt Zouma's impending departure for Saudi Arabia has forced the Premier League side back into the market.
West Ham technical director Tim Steidten has travelled to France to try and conclude negotiations after the centre-back's proposed move to Juventus stalled.
He has now stepped in with a counter offer that is understood to be a season-long loan with obligation to pay £34m next year plus a sell-on clause.
Talks are ongoing between the two sides with West Ham hoping to push through the signing over the weekend, but they will have to convince Todibo it is the best move for his career.
Meanwhile, Zouma is heading to Saudi Pro League side Shabab Al Ahli and manager Julen Lopetegui is now facing a defensive crisis with the new Premier League season just a week away.
West Ham have also considered Chelsea's Trevoh Chalobah and Diego Carlos of Aston Villa, but Todibo was their prime target - despite rejecting their overtures earlier in the window.
Todibo had set his heart on a move to Juventus but the Italians have been unable to reach agreement as they face financial difficulties of their own and have put eight first-team stars up for sale.