Rodrygo was substituted in the 20th minute of the La Liga contest, before Vazquez was replaced at the break, with both players picking up muscular issues.
According to transfer expert Fabrizio Romano, Rodrygo will be out until the beginning of December, while Vazquez is also facing three weeks on the treatment table.
Real Madrid beat Osasuna 4-0 on Saturday, with Jude Bellingham on the scoresheet, while Vinicius Junior hit a hat trick, with the Spanish champions picking up a valuable three points.
However, the victory came at a cost, as Los Blancos also confirmed after the clash with Osasuna that Eder Militao had suffered an ACL injury.
Rodrygo, Vazquez 'out until early December'
Militao will be out for the remainder of the season, with the centre-back potentially returning towards the end next year, depending on how his recovery progresses in the coming months.
Raul Asencio was introduced in place of Militao on Saturday, and the Spaniard could now be set for a run in the side, as David Alaba is also still out with an ACL injury.
Rodrygo and Vazquez are expected to be absent for Real Madrid's next three matches in all competitions, which come against Leganes, Liverpool and Getafe.
The duo could potentially return against Athletic Bilbao in early December, although they might be held back for the clash with Girona on the weekend of December 7.
How many Real Madrid players are currently injured?
Three more have been added to the treatment room, making the number of injured Real Madrid players seven, with Aurelien Tchouameni, Thibaut Courtois, Alaba and Dani Carvajal also out.
Tchouameni is expected to be on the sidelines for the next month with his ankle injury, but Courtois could be back before the end of the month, with the goalkeeper in the latter stages of his recovery from a muscular problem.
Carvajal is out for the campaign with an ACL injury, while it is still unclear when Alaba will be able to return to action.
Alaba has been missing since last December due to an ACL issue of his own, and it now seems unlikely that he will return to the field this year, having had setbacks in his recovery process.