Greening, 30, has been unable to recover from a toe injury and is expected to make an official announcement about his future tomorrow.
The first-choice hooker, Raphael Ibanez, has a rib injury that kept him out of last weekend's Powergen Cup win at London Irish. It leaves the Wycombe-based club with the 24-year-old Ben Gotting in that position.
Dowd, speaking on Wasps' clubcall line, the Buzzline, said: "Phil is a huge loss to the team, and we can't go through the season without replacing him. We will seriously have to look at adding another player to the squad. We have to have cover there both for A team games and the Premiership team."
Dowd, who propped for the All Blacks during his 60-cap Test career, also attacked Saracens' decision to play seven out of eight first-team forwards in Monday night's Guinness A League match at Henley.
Wasps lost the game, which finished after an hour when their flanker Garth Chamberlain was taken to hospital after sustaining a neck injury. Although the player was subsequently given the medical all-clear, Dowd was angry that Saracens had put experienced players up against a team of rookies.
The Saracens head coach, Mike Ford, explained: "After the last couple of results we felt that some of the players needed some extra game time, and through playing some of the guys for 40 minutes we were able to put them in a live situation which we just can't recreate in training."
But Dowd felt that Saracens had overlooked the purpose of the A League competition.
"Steve Diamond [the Saracens director of rugby] was angry with how his first XV had played on Saturday, but we have an A team side several of whom are still at school.
"It should have been a mismatch, but those Wasps boys didn't let that happen, and I think we got far more out of that game than they did. However, I think they should really look at the safety issues of putting out a pack full of internationals against a group of young lads. This league is all about developing young players," Dowd said.
One experienced player who is poised to return to action is Gloucester and England prop Phil Vickery. He has recovered from the shoulder injury he picked up last month and is expected to be available for Sunday's Powergen Cup match at Bristol.
Leicester last night announced they have appointed Sir Digby Jones, the director-general of the Confederation of British Industry, to the club's board. Sir Digby, a Tigers season-ticket holder, takes up his position on 1 November.
Sir Digby will replace David Jones, the Next plc chairman, who has served for eight years as a Tigers director.
"Leicester is a club with a great heritage and tradition and I am honoured to accept this invitation," Sir Digby said.
The Tigers chairman, Peter Tom, added: "Sir Digby brings with him a wealth of business knowledge and commercial expertise, which will be invaluable as we look to further strengthen our position as the leading rugby club in Europe and embark on an ambitious stadium development programme."
* Wales coaches have admitted that fly-half Stephen Jones is in a race against time to play against New Zealand on 5 November. Scott Johnson, the assistant national coach, said it would be "touch and go" as Jones recovers from an ankle injury suffered while playing for French club Clermont Auvergne on Saturday.Reuse content