Firestore Rules verify timestamp with a Flutter client

You can use the Timestamp to add constraints to the time field (docs).

Here is an example of how to ensure that the change was within a certain amount of seconds:

function withinSeconds(secs) {
  return request.resource.data.TIMEFIELD.seconds() - request.time.seconds() <= secs
    && request.resource.data.TIMEFIELD.seconds() - request.time.seconds() >= -secs
}

Edit

The above is for setting the value within a threshold of the request.time.

You can also just use the REST API in the mean time. Just make a write request that includes an update and a transform. The transform is where you would set the server timestamp. Here is a tool to help understand how to build the requests.

This has been implemented into the Flutter plugin for Cloud Firestore:

FieldValue.serverTimestamp()

Using this as a field's value will assign a timestamp equal to request.time to the field, server-side.

You can find out more about it in the API reference for cloud_firestore.