Oliver Andrews visited Marrakech in November and was arrested after being accused of using counterfeit money.
The 29-year-old’s family said he denied the charges and had received no help from the British authorities.
The Foreign, Commonwealth and Development Office (FCDO) said it had provided assistance.
The BBC has asked the Consulate General of the Kingdom of Morocco for comment.
On 10 November 2022, Mr Andrews, from Bournemouth, and a friend went to a nightclub on the last night of their holiday.
The next morning, Moroccan police arrested them and Mr Andrews was told that half of his money was counterfeit, according to his family.
The family said the two men were not given an official translator or the opportunity to speak to a lawyer.
They also said the men were “pressured into signing foreign paperwork”.
Speaking via the family, Mr Andrew’s solicitor, in Morocco, said both men had been charged with possessing and distributing counterfeit money within Morocco, and one count of creating an organised criminal group.
Mr Andrews was diagnosed with a heart condition in 2021 and requires daily medication, his family said.
They added he was not given access to his medication until his solicitor organised access on 16 December.
“The British Embassy in Morocco and the FCDO have refused to get involved,” they continued.
They said the British Embassy to Morocco and the FCDO were informed about Mr Andrew’s heart condition and medical requirements.
When Mr Andrews was visited by his solicitor the family said “he was in a very deteriorating mental and physical state”.
They added that, because he had not taking his medication, “his blood circulation was slowing down”.
The family also said they sent money to the embassy for Mr Andrews to make phone calls but added “this money was not given to him, to date”.
Alanna Cornick, Mr Andrew’s partner, said: “We just want to the embassy to do their job and go and visit him, and make sure to check on his welfare.”
“The condition that he’s living in is just heart-breaking on a daily basis,” she added.
She described how he had been kept in “cramped spaces” with 32 people in a 12-man cell, with no bedding or fresh air.
“I have had phone calls from him where he has been really really down and basically shared suicidal thoughts,” she continued.
“It is the worst because there’s nothing you can do.”
Zoe, Mr Andrew’s mother, said the family “have given up trying to get anything done through the embassy”.
“As a family, we feel very let down and deserted by our own government,” she added.
“We, under no circumstances, want help with the legal case, just solely his welfare.”
An FCDO spokesperson contested what the family said and told the BBC: “We are providing consular assistance to two British nationals arrested in Morocco and are in contact with the local authorities.”