The ex-Welsh under-20 internationals have been training with the senior squad in recent weeks, the Welsh Rugby Union said.
Ospreys hooker Lake and Dragons back row forward Basham have now been formally added to head coach Wayne Pivac’s group. The WRU added that no players have been released from the squad.
Pivac originally selected 38 players for the Six Nations, which Wales began with a 42-0 victory over Italy last weekend. Cardiff Blues wing Owen Lane, though, was released last month after suffering a hamstring injury.
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Lane’s Blues colleague Josh Navidi, who remains part of the squad, is currently sidelined because of hamstring trouble, while Saracens back Liam Williams has not yet recovered from an ankle injury that ended his World Cup in semi-final week.
But Dragons hooker Elliot Dee, Ospreys centre Owen Watkin and Scarlets scrum-half Gareth Davies are all expected to be in the selection mix for Saturday’s appointment with Ireland in Dublin after resuming full training following injuries.
– Press Association
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