Seven people have tested negative for swine flu after being in contact with the first Britons to contract the virus, the BBC understands.
The number of suspected swine flu cases in Scotland now stands at 32, the Scottish Government has confirmed.
The results are still pending on the other two people who had come into contact with Iain and Dawn Askham.
Health Secretary Nicola Sturgeon said tests in 15 other cases had returned negative results for the virus.
The newly-weds, from Polmont near Falkirk, were the first Britons to be diagnosed with swine flu.
She told MSPs the total number included 24 new suspected cases, all of them people with travel connections to Mexico or other affected areas.
There have been three more confirmed cases in the UK - two adults in London and Birmingham and a girl in Devon.
Ms Sturgeon stressed the risk to the public was "low" and officials were working hard to keep it that way.
All three of the new confirmed cases had recently travelled from Mexico.
Iain and Dawn Askham, the newly-weds from Polmont near Falkirk, were the first Britons to be diagnosed with swine flu.
Prime Minister Gordon Brown said they had mild symptoms and were responding well to treatment.
Ms Sturgeon said they were continuing to recover in hospital, adding that eight of the nine people who had been in close contact with the couple and had displayed mild symptoms had tested negative for swine flu.
The Askhams were admitted to a Lanarkshire hospital at the weekend after honeymooning in Mexico, and were said to be making a good recovery.
Test results in the ninth case are still outstanding.
Ms Sturgeon said: "The test results suggest that, to date, as far as we know, we have managed to prevent the spread of infection within Scotland.
The results of another 16 suspected swine flu cases in Scotland are still awaited.
"That, at this stage, is encouraging."
Of these, 14 people have travel links to Mexico or affected parts of the United States, and have shown mild symptoms of the virus.
The 23 cases under investigation were in Greater Glasgow and Clyde, Lothian, Forth Valley, Grampian, Tayside and Lanarkshire.
They included Edinburgh newly-weds Pete and Jenny Marshall, who have been quarantined at their home with friend Gemma O'Brien while tests are carried out.
NHS Grampian reported six suspected cases in its area.
According to NHS 24, helpline centres were taking 23% more calls than usual, with people seeking reassurance and advice, while daily website visits had risen by 70%.
Medical director Dr George Crooks said the service received 3,200 phone calls on Tuesday, with 10% of the inquiries relating to swine flu.
"However, only a small number of these calls relate to people with flu-like symptoms who have returned from affected areas abroad and many of these symptoms are common problems often experienced by returning travellers," he said.
"NHS 24 has increased its staffing resource and the service is performing extremely well with the rise in call demand."
In the next few days, leaflets with information about swine flu will be sent to every household in the UK.
Organisations are being warned to make their own preparations for a possible pandemic.
Meanwhile, the global outbreak has spread further, with the first death outside Mexico confirmed in the United States.
US health officials revealed that a child had died of swine flu in Texas.
About 160 deaths in Mexico are being linked to the virus, though only a handful of those have been confirmed.
Samples of the new strain of swine flu found in Mexico and the US are being sent to Britain where work will start on analysing the virus.
Leading bacteriologist Professor Hugh Pennington said the test results from Scotland would help answer how transmissible the virus was from person to person.
The World Health Organisation has raised its alert level to four - two steps short of declaring a pandemic.
