If illegal immigrants have a child on US soil, then that child is an US citizen due to the jus soli citizenship concept of the United States (see US constitution, 14th Amendment, Section 1). The current legal practice in that case is that the parents can then stay in the country so they can take care of the young US citizen. There are three fundamental rights which play a role here and which are currently prioritized as follows:
- The right of the child to live in the United States
- The right of the child to live with their parents
- The right of society to deport illegal immigrants.
No rights are absolute, and legislations always need to balance different rights against each other to find a compromise. One could also suggest a different priority. Who says that the rights of a single US citizen weight higher than the right of 320 million US citizens to not having to tolerate two non-US citizens living among them?
If one would balance them in the opposite order (3 > 2 > 1), then the logical solution of this order of priorities would be to deport the whole family. The child would still be an US citizen, though, and could return to the United States when they are old enough.
One could also propose the priority order 1 > 3 > 2. The child would then have to decide between leaving the US with their parents or staying in the US and living with foster-parents. Because the child is a minor, that decision would be made by their parents for them. Again, if they decide to leave with the child, the child can return later.
The US Supreme Court is the last instance to decide if balance decisions between different rights are fair and constitutional. I would expect that either such a legislation would soon land at the Supreme Court. I am not a US Supreme Court Justice, so I don't consider myself qualified to predict their decision.
Another option would be to change the 14th amendment and replace jus soli (citizenship by location of birth) by jus sanguine (citizenship by citizenship of parents). In that case the child of illegal immigrants would also be an illegal immigrant. But changing the US constitution is outside of the authority of the US president as it requires not just the federal senate and congress to agree but also the majority of state-governments to agree.