She has been detained in Iran for more than five years on spying charges after being arrested there in 2016 while taking her daughter to see her family.
Tulip Siddiq said Mrs Zaghari-Ratcliffe was still at her family home in Tehran.
Boris Johnson said it would not be sensible to comment “until we’ve got a final result” but said “delicate discussions are going on”.
Ms Siddiq, the Labour MP for Hampstead and Kilburn, said she understood British negotiators were in the Iranian capital.
Mr Johnson said he did not want to “tempt fate” and said that negotiations about “all our difficult consular cases have been going on for a long time”.
The prime minister added: “Everybody wants Nazanin home, we’ve been working on that for a long, long time, I do not want to do anything to interrupt conversations right now.”
According to Mrs Zaghari-Ratcliffe’s employer, the Thomson Reuters Foundation, her lawyer Hojjat Kermani, when asked whether she would be released, said: “I am hopeful that we will have good news soon.”
A £400m debt relating to a cancelled order for 1,500 Chieftain tanks dating back to the 1970s had been linked to the continued detention of Mrs Zaghari-Ratcliffe and other UK-Iranian dual nationals held in the country – although the government has said the two issues should not be linked.
Downing Street said it was committed to paying the debt and was “exploring options to resolve it”, but said it had not been resolved.
A Foreign Office spokesman said it continued to “explore options” to resolve the debt but would not comment further as discussions were ongoing.
He added the Foreign Office would not comment on speculation about Mrs Zaghari-Ratcliffe but had long called for the release of “unfairly detained British nationals in Iran”.
Mrs Zaghari-Ratcliffe, 43, who has always denied the charges against her, was first jailed for five years in 2016 after being accused of plotting against the regime – spending the last year of her sentence under house arrest at her parents’ home.
After that sentence expired she was then sentenced to another year’s confinement in April 2021 on charges of “spreading propaganda”, which has been served at her parents’ house in Tehran.
Her husband Richard Ratcliffe, who lives with their daughter Gabriella in Hampstead, London, has campaigned for her release including by going on hunger strike in October last year.
Mrs Zaghari-Ratcliffe’s sister-in-law Rebecca Ratcliffe told BBC Radio 5 Live the media reports seemed “like a good positive sign” but said there had been “so many false hopes” over the last five or six years so it was hard to tell “if this is a really positive sign or just the Iranian government playing games again”.
She said her sister-in-law was on edge when things like this happened because she does not want to get her hopes up.
“We have everything crossed but remain sceptical,” she said.
Her mother-in-law Barbara Ratcliffe told the BBC Mrs Zaghari-Ratcliffe “seemed really quite upbeat when I last spoke to her”, but added that the family was “all a bit battle-scarred” having had disappointments in the past.
Sacha Deshmukh, Amnesty International UK’s chief executive, warned the latest reports should be treated with caution as there had been “false dawn after false dawn” in the long-running process.
Rupert Skilbeck, director of the Redress human rights organisation, also echoed the words of caution. “So we remain cautious and continue to encourage the UK government to do the right thing and ensure the debt it owes to Iran is paid, and that it does everything in its power to secure Nazanin’s release,” he said.