on the top of the engine drivers side near the brake fluid resevior there is a bracket that is attacthed to the firewall and a rubber mount that connects to a braket on the engine, look at the rubber mount in the bracket, it is black and round with a bolt going through it.

If you can see the mount is worn or missing some of the rubber it needs to be replaced, when mine failed i could see right through it from one side to the other. it should be solid and stiff.

The mount will wear out and cause excessive movement of the engine. It is pressed in and is very inexpensive to replace.

Also take a look at the fan belt and see if it looks worn, the moan may also be a bearing on a pully going bad (alternator, power steering, etc.) they will moan and make strange noises.

Since the noise continues after the motor is running you can rule out the starter.

Dan