The 16-year-old was seen on CCTV at Union Square shopping centre in Aberdeen last Saturday afternoon, before boarding a Stagecoach Bus bound for Banchory around 20 miles away.
It is thought he bought a ticket for Raemoir Road in Banchory, but a passenger has told police they think Liam got off at Crathes.
There have been no confirmed sightings since, and concern is growing for the 16-year-old with police divers searching the River Dee on Friday.
A week on from the last sighting, former Aberdeen player McLeish recorded a video message urging football fan Liam or anyone with information on his whereabouts to contact police.
The Scotland manager said: “This is an urgent appeal for anybody who knows the whereabouts of Liam Smith.
“Liam, if you’re watching this message, please come home. Your mum, Alix, and your family are very anxious to see you returning.
“Liam went missing on November 17 so anybody who has seen Liam in these days since that date please contact Police Scotland or his family.”
Officers are returning to “key sites” in Union Square, Crathes and Banchory on Saturday to speak to people who are regularly in the areas and may have seen the teenager last week.
Liam, from the city’s Queens Road area of Aberdeen, is described as white, 6ft 4in and slim, with short brown hair.
He was wearing a dark top, blue jeans and Nike Air Max trainers, and was carrying a camouflage print North Face bag.
Earlier this week, his mother Alix Smith said: “As each day passes we are becoming more and more desperate to find Liam.
“We are literally clinging to the hope that he is out there somewhere taking shelter.
“Not knowing where Liam is is just so painful for us and we just want to find him safe.”
Chief Inspector Martin Mackay said the force has received around 100 calls with potential information.
He said: “It is one week today since Liam went missing and I don’t think it would be wrong for me to say his disappearance has affected not just his family and friends but the entire north east community and beyond.
“If you have information call 101 quoting ref 1302 of November 18.”