Trachycarpus is *not* a suckering palm. It develops a solitary trunk
with a single growing point except in extremely rare circumstances
where non-fatal damage to the growing point causes fasciation
resulting in a multi-headed plant. Occasionally, several seedlings
are planted together, giving the impression that the palm is
suckering, but separating them once established is nigh-on impossible
without incurring fatal damage to the individual plants.

If the palm in question is genuinely producing offsets, then it is
almost certainly Chamaerops humilis; many forms of which produce them
very freely. Unfortunately, propagation of Chamaerops from offsets is
a painfully slow process and the failure rate is usually very high.
Full root formation even with basal heat takes several years and if
you finally get a rooted plant, it will not be any larger than one
grown from seed sown at the same time as the offset removal.
Chamaerops is such an inexpensive palm that it is far cheaper and
easier to buy a new plant.