She said some people were obtaining fake passports from neighbouring countries where, unlike Zimbabwe, citizens did not need UK entry visas.
But some Foreign Office sources have dismissed her words as alarmist, BBC political editor Nick Robinson said.
It is thought more than 700 people have died in Zimbabwe’s cholera outbreak.
Ms Smith is thought to have told cabinet colleagues that people fleeing Zimbabwe had bought false passports in countries such as South Africa and Botswana.
But asked whether the situation in Zimbabwe could lead to a refugee crisis in the UK, Foreign Office minister Lord Malloch Brown said: "I think crisis is too strong a word for it.
"She [Jacqui Smith] wants to make sure that her border agency is properly prepared for this.
"She briefed the cabinet on that way, prior to a longer discussion on Zimbabwe."
But our correspondent said the crisis in Zimbabwe was "causing real tension inside the government".
At the moment Zimbabweans arriving in the UK need a visa, while many travelling from the rest of southern Africa do not.
Ms Smith wants that to change that, having been told that a growing number of refugees are entering the UK using false papers.
Zimbabwean President Robert Mugabe said on Thursday that the cholera outbreak had been contained.
However, the UN Office for the Co-ordination of Humanitarian Affairs said the toll from the disease had risen to 783 and that 16,403 people were believed to have been infected.
Save the Children in Harare said these figures were an underestimate and the epidemic was almost certainly worsening. SOURCE: BBC