The warning comes as the Procurement Reform (Scotland) Bill, which aims to make the awarding of public contracts more consistent and transparent, is debated by MSPs today.
Unison said yesterday much more must be done to "stop tax dodging companies winning public contracts".
It called for the Scottish Government to consider adding provisions to address "aggressive" tax avoidance as well as illegal tax evasion in the procurement process.
The union says it is also "disappointed that the Scottish Government made no mention of the living wage in the Bill or any supporting documents". It wants a requirement that all contracting authorities stipulate payment of the living wage as a condition firms must meet.
Dave Watson, head of bargaining and campaigns, said: "It is wrong that companies who dodge paying their fair share of tax should be awarded public contracts. Almost £10 billion of public money, over one-quarter of the Scottish Government budget, is spent on public procurement. It is an important lever to change the behaviour of big companies who do not play by the rules."
Susan Torrance, Scottish Federation of Housing Associations policy manager, said its members need to be able to make "sensible" decisions based on their size and scale.