In the overnight attacks in Santiago, the capital and largest city where the pope will arrive Monday, the churches were hit with firebombs and then sprayed with accelerant.
"The next bombs will be in your cassock," read pamphlets found outside one of the churches.
A bomb squad was also deployed to a fourth church, where a barrel of flammable liquid was believed to be inside.
The pamphlets exhorted the cause of the Mapuche indigenous people, who are pushing for a return of ancestral lands and other rights. Francis will celebrate Mass and meet with Mapuches in the southern city of Temuco on Wednesday.
After a security meeting, Ms Bachelet said the Andean nation of 17 million was prepared for the first papal visit since St John Paul II came in 1987.
"I also want to invite you all to experience this visit in a climate of respect, solidarity and happiness," Ms Bachelet said.
There were no immediate arrests over the firebombings, and authorities downplayed their significance with Interior Ministry official Mahmud Aleuy calling the damage "minor".
Chilean police did not immediately respond to queries about whether new security measures would be taken after the attacks.
Earlier this week police said 18,000 officers would be deployed during Francis' visits to Santiago, Temuco and the northern city of Iquique. Police will also have helicopters on hand and monitor events with drones.
It was unclear who might have been behind Friday's attacks. A small minority of Mapuches have used violence to further their cause, and in recent years churches have been targeted.
Chile also has a handful of anarchist groups that periodically attack property and clash with police during protests.
The pamphlet that threatened the pope mentioned the Mapuche cause and called for the liberation of "all political prisoners in the world".
Francis' visit to Chile and Peru aims to highlight immigration, the suffering of indigenous peoples and protecting the Amazon rainforest.
