Giuseppe Profiti, the former president of the hospital's foundation, and the foundation's former treasurer, Massimo Spina, have been charged with misappropriation of funds.
But Cardinal Bertone, who was Vatican Secretary of State under the former Pope, Benedict XVI, has denied any involvement.
Tweet: "Penthouse of Bertone, Cardinal defends himself and says 'he never used money of the Bambino Gesu Foundation'"
Mr. Profiti and Mr. Spina appeared at a preliminary hearing on Tuesday (July 18), and their trial is due to begin in September.
A veteran Italian journalist, Franca Giansoldati, reported recently that Cardinal Bertone had given a memo to investigators last November.
The scandal is the latest to hit the Vatican.
Pope Francis is trying to reform the city state, but his auditor general, Libero Milone, left his post recently without explanation, and the man he picked to be Economy Secretary, Cardinal George Pell, resigned to return to Australia to fight sex abuse charges.
Earlier this month, Italian police raided an apartment used by a priest who worked for Cardinal Francesco Coccopalmerio, where a drug-fueled gay orgy was taking place.
Cardinal Bertone said any suggestion he was involved in the misappropriation of money was "scandalous" and "libelous."
Mr. Profiti has reportedly argued that it was justified to use the foundation's money to refurbish Cardinal Bertone's apartment because fundraising receptions for the hospital were often held there.
The Italian press has claimed the contract was awarded to Italian businessman, Gianantonio Bandera, whose construction company went bust, leaving the work uncompleted.
Mr. Bandera is also alleged to have used an English sub-contractor.
He and the president of thec Foundation, Mariella Enoc, are expected to testify at the trial, which will resume while Pope Francis is visiting Colombia.
When the scandal first came to light, Cardinal Bertone reportedly made a voluntary US$170,000 donation to the Bambino Gesu Foundation.
The affair has also highlighted tensions between Cardinal Bertone and Cardinal Giuseppe Bertello, head of the Government of the Vatican City State.
Cardinal Bertello was apparently aggrieved at not getting the post of prefect of the Congregation for the Evangelization of Peoples, the Vatican's missionary department.