I am not aware of the truth about the Islamic financial institutions borrowing from conventional banks/lenders on interest but I have heard the same concern from many other people as well.
If it is true that Islamic financial institutions are borrowing money from conventional banks/lenders on interest, and then using the same money to provide financing to consumers, then they are guilty of participating in interest based transaction and they should figure out a way to find investors who can share a profit in real partnership instead of borrowing money on interest terms.
Now, if a consumer made a separate transaction with such financial institution and the transaction between the consumer and Islamic financial institution is free from Riba in every way then this transaction will be considered Halal and permissible.
The reason why this transaction will be considered Halal and permissible and lawful is because this is a separate transaction. The financial institution will be responsible for its dealing in Riba and they are held accountable to Allah for that but the transaction that takes place between the consumer and financial institution would be permissible as long as it is free from any direct involvement of Riba.
However, Muslim consumer should voice their opinion of displeasure to the executives and decision makers at these Islamic financial institutions so they can work hard to find investors who would deal with them while complying with the teachings of Islamic Shari'ah.
And Allah knows best.
Mufti Ikram ul Haq
Fatwa Center of America