Ms Thomson withdrew from the SNP whip in 2015 when a police inquiry into allegations of mortgage fraud was launched but said it was not her choice and that she received no support from the party hierarchy.
The former Edinburgh West MP was told last month that she will not face court proceedings and asked for an apology from SNP leader Nicola Sturgeon.
Mr Salmond, who stood down as leader in 2014, was a Westminster colleague of Ms Thomson and said he had been a “strong supporter of her”. He also wants to see her return to the party.
Ms Thomson was one of five people named in a report sent to prosecutors last December following the police investigation but Scotland’s Crown Office said there would be no criminal proceedings due to an absence of sufficient credible and reliable evidence.
Mr Salmond said: “It’s a difficult position for any political party to be in.
“It was an extremely difficult situation, which the SNP handled badly.
“I thought, whenever the parliamentary group voted on this, that you cannot have a situation where people are denied natural justice. There is a point where you have to take action and I happen to believe that’s at the point of charging, dependent on the offence.”
Ms Thomson, who sat in the House of Commons as an independent, stood down as an MP at the general election in June.
She was linked to deals involving Christopher Hales, a solicitor who was struck off for professional misconduct involving transactions in 2010 and 2011.
She said she was not aware Mr Hales had been struck off until it emerged in the press in 2015.
Mr Salmond took aim at the media over its reporting of the situation.
Speaking at a press conference ahead of his first show at the Edinburgh Festival, he said: “Although I think the SNP made a mistake, they weren’t the instigators of the mistake.
“The people who bear the heaviest burden of responsibility for the position that Michelle is in now and has been over the last two years are your publications.
“That should be reflected upon because (the media) have effectively removed the political career of someone I thought was an incredibly talented female politician and that is greatly to be regretted.”
Ms Thomson said “she very much welcomes the ongoing support” of Mr Salmond and is looking forward to a “frank meeting” with the First Minister “as soon as possible”.