The Scottish Defence League (SDL), an offshoot of its English counterpart, is planning to hold a protest in the city on Saturday.
Similar demonstrations in England and Wales have seen angry confrontations with opponents and police.
A group called Scotland United, supported by politicians, trade unionists and faith groups, has organised a rally in Glasgow on Saturday to oppose the SDL demo.
Deputy First Minister Nicola Sturgeon is among the speakers at the rally, which will be followed by a march through the city centre and will culminate in a minute silence at George Square for victims of racial violence.
Strathclyde Police assistant chief constable Fiona Taylor said they were "fully prepared" for the events.
She said: "A significant amount of planning and preparation has been undertaken by the force and a detailed policing plan is in place with sufficient resources available to respond to any incidents.
"We would like to reassure members of the public who intend to come into the city on Saturday that our aim is to keep disruption to a minimum. However, we would suggest that people allow extra time for their journey.
"As always, public safety is paramount for the police, and should any event turn out to pose a risk to public safety then we will respond accordingly."
The time and location of the SDL protest is not yet known.
The English Defence League started in June and has held several demonstrations, including events in London, Manchester and Leeds.
The group describes accusations it is racist or fascist as "flat out untrue".
The SDL was formed in protest against Muslim "extremists" and "jihadists".
They say the UK Government is not taking significant action against the "Islamist assault" which they claim is threatening British culture.
Scotland United, whose organisers include human rights lawyer Aamer Anwar, said it will celebrate multi-cultural Glasgow in a gathering at Glasgow Green.
The group said that, following the racist murder of Indian naval officer Kunal Mohanty and the BNP's fourth place in the Glasgow North East by-election, people would be asking themselves what sort of Scotland they want.
A spokesman said: "It is vital that we do not allow these racist thugs to terrorise and intimidate Muslim and Asian people.
"We are determined to turn Saturday November 14 into a day that Scotland says loud and clear we do not want fascists on our streets."
Ms Sturgeon said: "We want all our citizens to play a part in shaping our nation's future articulating the hopes of a diverse, multi-cultural, modern Scotland.
"As the far-right Scottish Defence League prepares an anti-Islamic demonstration in Glasgow on Saturday, it is important for the whole nation to send a clear message condemning all forms of bigotry and discrimination.
"To show opposition to that demonstration, like thousands of others, I will attend the Scotland United rally in Glasgow to stand shoulder to shoulder with Scotland's Muslim communities."
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