I have an older 21" self propelled mower with a push button starter and pull rope starter. Neither will start engine.I

You need to remove the starting mechanism and make sure that there are no kinks, etc. in the starter rope.

Been there, but not 100% sure why. If machine also has self propel wheel drive then try turning the motor over by pushing mower at same time as pulling starter rope or disengaging self propel by releasing handle quickly a couple times.

Remove spark plug and observe condition. Fouled with oil, bad motor. With plug removed pull rope several times to blow out cylinder. Spray WD40 in spark plug hole, pull rope several times see if oil blows out. Spray a lot of starter fluid in spark plug hole, replace plug. Spray starter fluid in air cleaner area of carb, try to start. Replace spark plug, try to start. Ensure oil is full, all mowers have a low oil sensor now. 30w oil. Always use Stabile in gas.

Does the starter rope rewind? If So and You removed the engine shroud you have to be sure the Pull Starter lines up with the flywheel nut. If the rope does not rewind the return spring is broken so you have to get a spring and remove the shroud and install it. I would take the shroud off and with me to get the spring and the place may install it for you. Bolt the shroud back on and you should be in business.

starter clutch under recoil is seizing on shaft...need to polish shaft and lubricate....recoil housing will probably be damaged as well

The drive belt that pulls maybe stretched causing it not to pull as it did. These can be replaced.

You need to remove the starter cover and rewind the rope a bit.

It is actually not too hard to do. And you dont need any parts!

You will find the retractable spring is broken you can go to any lawn mower shop and buy a replace me, also Wal mart and others you will have to wind it up till rope is tight then hook to the flywheel where the slot is. cost about $10.00 also lube the center hub bearing with wd 40

It's possible the rope came out of it's pulley groove and is jammed inside the engine housing.

the starter clutch probably needs cleaning. It is located right on top of the fly wheel. Pull the the pull rope housing off and you can see it