Big money offers have reportedly been made to Lawes, who is in his 17th year at Saints, by Provence and Brive, while fellow ProD2 promotion hopefuls Beziers are also said to have entered the bidding.
Lawes, who turns 35 on Friday, had made no secret of the fact that money is the main motivating factor in what will likely be his last professional contract.
Having announced a £1.1 million loss before tax in the year to June 2023, Saints are focused on living within their means and won't be in a position to match those offers from across the channel.
However, the emotional pull of leaving the only club he has ever played for might still count in Saints' favour: Dowson says he very much hopes that the situation will be resolved one way or the other well before the Premiership leaders attempt to pick up where they left off in the first game back after the long break, at Bristol on March 22.
"I don't think it is a million miles away but there is nothing absolute. We are still hoping we can do something but we'll wait and see on that," he said.
Despite protestations from the majority of clubs, Dowson confirmed that next season's salary cap would revert back to the pre-Covid figure of £6.4million.
There needed to be a unanimous 'no' from all 10 Premiership clubs for the salary cap to remain at a more sustainable £5million, but the likes of Saracens and Bristol chose not to read the room and voted to increase it.
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