Mahmoud said the 20 men held on Friday included Mahmoud Hussein, a member of the small Guidance Office, which acts as an executive committee for the organisation.
The leaders gathered on Friday evening at the home of Nabil Mukbil, a businessman from the west Cairo suburb of Muhandiseen.
Most of the men taken were from Cairo, but seven came from more distant provinces of Egypt, Mahmoud said.
A spokesman for Egypt's interior ministry had no comment on reports of the arrests.
The members allegedly stand accused of belonging to an banned organisation, possession of illegal documents, and holding a meeting to plan illicit activities, according to an AFP report.
The group, which published a list of those arrested on its website, said that the gathering in question was only "a routine meeting."
Mohamed Habib, the deputy leader of the Brotherhood, said that the crackdown was timed to harass the group as it prepares to announce the programme of a political party which it would like to establish on September 1.
Egyptian law bans parties based on religion, a ban now enshrined in a constitutional amendment passed this year.
Around 400 members of the Brotherhood are now in detention, most of them without charges or trial, after a crackdown which began in December last year.
On Friday, Erian, who has been in and out of detention for more than 15 years as a result of his involvement with the Brotherhood, allegedly missed out on a holiday in Turkey because immigration officials at Cairo airport stopped him from travelling.
It was the third time in four months that Erian, a medical doctor, has tried and failed to pass through Cairo airport to travel abroad.
He was part of a group of 27 doctors travelling to Turkey on Friday morning on a vacation organised by the Doctors Syndicate, of which Erian has been a senior official.
He said the syndicate had given the authorities a list of the doctors in the group before travelling and no one had objected to his name.
"I no longer ask [if they will let me travel] because it has become silly, that one should have to ask for one's natural right," he added.