Ludlam, who is set to be out of contract at the end of the season, has been linked with Toulon, with the French media reporting that a three-year deal is on the table from the Top 14 giants.
Saints have already seen Dan Biggar and David Ribbans move to Toulon in recent times.
But Dowson insists nothing has been decided with Ludlam as yet.
"We are desperate to keep Luds," the Saints boss said. "He's club captain and he's exactly what the club's all about in terms of someone coming through the Academy and working incredibly hard on his game in getting to the very top level.
"He's a top player and off the field, with his leadership qualities, we're desperate to keep Luds.
"We're in negotiations at the moment and, as it stands, nothing has been formalised so we're going to keep pushing on that to keep him here.”
When asked whether he is resigned to losing Ludlam, Dowson said: "I'm never resigned to losing him.
"We had this conversation last year with Courtney (Lawes) and clearly there's lots and lots of things at play here so it's a complicated situation.
"But first thing I want to say is that Lewis Ludlam, we love him and we want to keep him, but we're aware that there are other clubs who see that quality and want to take him as well."