The Madrid-based paper has claimed that the midfielder mulled over the decision last night, and has asked his agent Inaki Ibanez to begin negotiations on a new contract.
Los Blancos have attempted to tie the Spain international to a new deal for almost six months now – with his current contract set to end in June 2014.
Reports on Friday suggested that the former Liverpool man had decided to leave Santiago Bernabeu.
But Marca have insisted on Saturday that Alonso will now sit down with President Florentino Perez and discuss terms.
Talks on a new deal won't begin until after January 6.
The issues up for discussion will be the length of the contract and his wage packet. The 32-year-old will want a minimum two-year extension, with a possible optional third year related to his performances and fitness. As for the financials, a salary of around €7 million a season is likely.
The sports daily has also revealed quotes attributed to Cristiano Ronaldo, telling Los Blancos to sign his team-mate Alonso as soon as possible.
“Xavi is irreplaceable,” he is believed to have said. “There is no one else like him.”