The England striker damaged his right knee during Saturday's defeat at Manchester City, and Southampton manager Mauricio Pochettino afterwards that the injury was "not looking good".
An update on the severity of the damage was expected by Monday, but swelling in the joint means he has not yet been scanned.
Rodriguez's Southampton team-mates have struck a sombre note since the incident, and Jack Cork has followed Rickie Lambert in offering his sympathies.
Speaking to the Daily Echo, Cork said: "It's horrible seeing your close friend who's doing so well, and got such a great chance of going to the World Cup, get a bad injury like that, and I just hope his recovery goes well.
"I've seen him from Burnley to here and the way he's developed, and I see how hard he works, and to just have it taken away from him like that is just horrible, and he's so close to going to the World Cup. My thoughts are with him and all his family."
Rodriguez has scored 15 Premier League goals this season and made his England debut against Chile in the autumn.
Southampton midfielder Morgan Schneiderlin told the Daily Echo: "Listen, my point of view is he had the (World Cup) tickets already in his pocket and an injury was the only thing that could have taken the ticket out of his pocket.
"It's such a shame. I just hope the scan is going to say something nice, and I hope he's going to be back on the pitch as soon as possible."