There is no specific threat but officers will be working to reassure and engage with communities of all faiths, Police Scotland said.
First Minister Nicola Sturgeon expressed solidarity with the Muslim community after attackers opened fire at two mosques in Christchurch during Friday prayers.
The Muslim Council of Scotland said it was “time for us all to stand together against this hatred” and said a vigil would take place at the top of Buchanan Street in Glasgow at 5pm.
On a visit to Glasgow Central Mosque with Justice Secretary Humza Yousaf, Ms Sturgeon said: “This terrorist attack on Muslims in their place of worship was horrific and cowardly and first and foremost my thoughts are with the victims and their families, with New Zealand’s Muslim community and indeed with all of the people of New Zealand.
“New Zealand is on the other side of the world from Scotland but I know that what happened there feels very personal and close to home to our own Muslim communities which is why I wanted to come here to Glasgow Central Mosque today to show my solidarity and offer some reassurance.
“The awful events in New Zealand are a very stark reminder of the duty that all of us have to stand firmly against the hatred and the far right ideology that motivates attacks like this one.
“There is no place in of our societies for Islamophobia, anti-semitism or racism of any kind and it is important that we send that message strongly and in a very united way.
“The police were here as I visited the mosque today. I know they are anxious to reassure the Muslim community at mosques, not just here in Glasgow, but across Scotland.
“Glasgow Central Mosque is in my constituency, there are several mosques in my constituency. I know the relationship between the community and the police is already a very strong one, but the police are right to talk about stepping up their patrols in the wake of what happened.
“It is absolutely fundamental that anybody of any religion or faith feels safe at their place of worship and that’s something we must make sure is the case here in Scotland.”
Politicians across Scotland expressed their horror at the attacks and Holyrood Presiding Officer, Ken Macintosh, said the Parliament would be flying flags at half-mast.
Justice Secretary Humza Yousaf tweeted a picture of Glasgow Central Mosque and said: “Go (to) Friday prayers every week, today feels different, already talking to worshippers who feel shaken, who know this attack wasn’t in a vacuum but hatred has gone unchecked, even encouraged by those who should know better.”
Scottish Labour MSP Anas Sarwar said: “This terrorist attack is designed to divide our communities, but an attack on one is an attack on all.
“In the aftermath of this distressing tragedy, we must unite and work together to confront hatred. This is not a fight for one individual community, it’s a fight for all of us.”
Scottish Green Party co-convener Patrick Harvie wrote: “The world must unite against such vicious far right hate; tolerating its presence in our society and in our politics has become normalised, and must end.”
Scottish Secretary David Mundell wrote: “My heart goes out to the families of those caught up in this outrage and the people of New Zealand.”
Scottish Liberal Democrat leader Willie Rennie described the attack as “sickening”.
Deputy Chief Constable Will Kerr said: “We are monitoring events in New Zealand closely and send our condolences to all those affected.
“We stand together with all our communities and partners here and overseas, and will continue to work with them to counter the threat no matter where it comes from.
“Today we will be stepping up reassurance patrols around mosques and increasing engagement with communities of all faiths, giving advice on how people and places can protect themselves.”
Anyone wishing to report anything suspicious is asked to call the confidential anti-terrorist hotline on 0800 789 321, or in an emergency call 999.