Bleed air systems for use with aircraft and related methods are disclosed. An example apparatus includes a compressor having a compressor inlet and a compressor outlet. The compressor inlet to receive airflow from a first air supply source. An air mixing device having a first mixer inlet to receive compressed air from the compressor outlet and a second mixer inlet to receive bleed air from a bleed air system. The bleed air to provide a motive fluid to enable the air mixing device to mix the bleed air and the compressed air to produce mixed air for the anti-icing system.