She will tie up down river from Govan where she was launched and across from the Scotstoun yard where she was fitted out.
She is the second of a new generation fleet of six Type 45 destroyers which were built in the Glasgow yards owned by giant defence contractor BAE Systems.
Dauntless is due at the King George V Dock tomorrow and will remain in port until Monday.
The last Type 45 to visit the city was sister ship HMS Defender.
Defender is the only Type 45 officially twinned with Glasgow.