Yes, push is a Scrabble word!

Verb

1. Move with force, "He pushed the table into a corner"

2. Press, drive, or impel (someone) to action or completion of an action

3. Make publicity for; try to sell (a product)

4. Strive and make an effort to reach a goal

5. Press against forcefully without moving

6. Approach a certain age or speed

7. Exert oneself continuously, vigorously, or obtrusively to gain an end or engage in a crusade for a certain cause or person; be an advocate for

8. Sell or promote the sale of (illegal goods such as drugs)

9. Move strenuously and with effort

10. Make strenuous pushing movements during birth to expel the baby

Noun

1. The act of applying force in order to move something away

2. The force used in pushing

3. Enterprising or ambitious drive

4. An electrical switch operated by pressing

5. An effort to advance