The 25-year-old New Zealand international, who has signed a three-year contract with the Giants, missed the start of the new Super League season after being stranded in his homeland by red tape. However, he was granted his visa a week ago and is due to arrive at the John Smith's Stadium later in the day.
Ta'ai is unlikely to be considered for Sunday's home game against London Broncos but could be a handy replacement for O'Donnell, who looks set to cut short his stay in Super League. The 32-year-old back-row forward still has a year remaining on his contract but is wanted by Sydney Roosters in time for the start of the new NRL season next month.
Huddersfield had no comment but Roosters recruitment manager Peter O'Sullivan said: "We're interested if a lot of things could happen."
O'Donnell has made just 23 appearances in two years because of injury and underwent a shoulder reconstruction at the end of last season, but he made an impressive return in Huddersfield's opening 40-4 win at St Helens last Saturday.
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