But he insists talk of the club signing the Toulouse No.8 is merely ‘gossip and speculation’.
Last month, reports in France suggested that the 29-year-old was being courted by Saints, Saracens, Clermont Auvergne and Montpellier.
But during the past week, there have been suggestions that Northampton and Bath are the front-runners to snap up Picamoles, whose current Toulouse deal runs until June, 2016.
It has been mooted that the player has agreed to sign a lucrative deal with an Aviva Premiership club.
However, Bath have since distanced themselves from the player, with first-team coach Darren Edwards saying: “We’ll always look at good players and if we get the opportunity we would sign them, but there’s nothing in that.”
That appeared to leave Saints as a likely destination, especially as they lost star man Samu Manoa, who was a key figure at No.8, to Toulon during the summer.
The club declined to comment on the rumours on Tuesday, but when questioned at the weekly press session, King said: “I’ve read the same stories and we’re flattered to be linked with those sort of players, but it’s just gossip and speculation.”
Picamoles also spoke out on Wednesday in a bid to quell speculation.
He said: “There are a lot of things that are said now.
“Me, this week, I’m on vacation. I try first of all to digest this disappointment and end of the difficult World Cup.
“I will talk with my club, Stade Toulousain, next week.
“As for my future, I will meet my current president (Jean-René Bouscaltel) and Fabien Pelous (sporting director) next week. So until then, I will not do anything. “
“Everything that can be said about it will only be as pure speculation for now, I have made no decision. I’ll make a decision quickly. But for now, I take my family and the holiday. Before taking any decision whatsoever, I will meet my leaders next week and make my decision.
“All that has been said before cannot be true, since for now there is no decision that has been taken.”