A clear indication of how long the forward will be sidelined is hoped to arrive early this week.
But Saints boss Mauricio Pochettino confirmed that the immediate signs did not look good, with the reaction of many of the players after the 4-1 defeat at the Etihad Stadium suggesting they are also expecting bad news for the World Cup hopeful.
“That’s the worst part of the day,” said Cork, who feared Rodriguez’s absence would extend into “months rather than weeks”.
“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.”
Cork explained that the disappointment of losing to City paled into comparison with how everyone felt for Rodriguez, but Saints were nonetheless frustrated to have lost.
Having controlled much of the first half, the game changed when the hosts scored two goals in added time – the first of which was clearly offside – giving them a 3-1 lead at the break.
“We feel a little bit hard done by,” admitted Cork. “It’s disappointing. Some decisions changed the game and it was a bit out of our hands we felt.
“It was miles offside and, at that stage of the game, it’s horrible to have that happen.
“Against the bigger teams, things don’t really seem to go our way.
“We put in a great performance first half and to go 3-1 down was a bit strange.
“We went in at half-time and it just felt like there was nothing we could have done.
“No matter how well we played we couldn’t have changed it.”