Due on Sale
Nearly every mortgage agreement ever written contains a "due on sale" clause. The due on sale clause generally provides that if you ever transfer the mortgaged property before paying off the mortgage then the mortgage lender has the right to immediately demand full repayment of the outstanding mortgage loan balance. A sale typically means any type of transfer, including a transfer to children regardless of whether any money changes hands.
Other than the due on sale clause, there is no legal restriction against transferring property while a mortgage is attached to that property. Accordingly, you can try to alter the terms of your agreement with your mortgage lender. For a small fee of about $100 or $200, many mortgage lenders will waive the due on sale clause. If your mortgage lender approves a waiver of the due on sale clause then you are free to transfer your home to your children even while the mortgage remains outstanding. The mortgage will, however, remain attached to that property even though you no longer own it.
Assumption or Cosign
Lenders are probably not likely to waive the due on sale clause unless the new home owner agrees to be a cosigner, or to fully assume, the mortgage loan. In other words, your children may have to become additional borrowers under the mortgage loan. In that case, your children will have to sign contracts with the mortgage lender before you carry out the transfer.
If your lender will not agree to waive the due on sale clause then you might consider creating a living trust. Under the living trust you will define the terms and conditions for when the home should transfer out of the trust and to your children. Federal law prohibits lenders from enforcing a due on sale clause in regard to transfers to a living trust. You will need to identify in the living trust agreement exactly when your children will take title to the home. The easiest solution might be to have the trust hold title to the property for as long as the mortgage remains outstanding, with the home passing to your children immediately upon full payment of the mortgage.