Officers said they could not rule out a connection between the shooting of a 23-year-old in Bishopbriggs and that of a 36-year-old man in Springburn on Christmas Eve.
This follows the fatal shooting of 21-year-old Steven Pollock in Paisley on Wednesday night. He was found dead at a friend's house in Weaver Terrace, Paisley, at around 11.15pm. Detectives said there was no apparent motive for the attack.
Officers said they did not believe there was any link between the murder in Paisley and the shooting in Springburn. However, they are still looking at whether there could be a link between the shootings in Bishopbriggs and Springburn.
Prior to the incidents in the past week, senior police officers said guns had accounted for just four murders in the past two years. Gun crime is still relatively unusual and makes up just 1% of crime in Scotland. Most homicides north of the Border are caused by a knife or sharp instrument, such as a broken bottle.
Last year there were 643 firearms offences across the whole of Scotland. The total number of offences include 93 for imitation firearms. There were 13 attempted murders involving a firearm and just one murder in Strathclyde involving a gun in 2010/11.
The 23-year-old man who was shot in Bishopbriggs is being treated in hospital after he was targeted near a golf range at around 12.30pm on Saturday. Strathclyde Police said four people were spotted running from the scene towards a nearby retail park.
A police spokeswoman said: "Anyone witnessing the incident or who saw the males running from the scene should contact Baird Street CID."
It happened less than 12 hours after a 36-year-old man was shot at by the passenger of a car in Springburn.
The spokeswoman said: "About 2am on Christmas Eve a 36-year-old male was seriously injured after being shot at by the passenger of a car which had approached him as he walked down Galloway Street in Springburn. He was walking with a friend. He was taken by ambulance to the Glasgow Royal Infirmary. Hospital staff said he was in a serious but stable condition."
Police said another murder inquiry was under way after a man died in hospital from stab wounds.
Christopher Bergin, 42, was found seriously injured in the stairwell of flats at Millhall Court in Plains, near Airdrie, in the early hours of Friday morning.
Mr Bergin, from Plains, was taken to Monklands District General Hospital where he died from his injuries on Christmas Eve.