No. The current offer becomes the starting bid of the no reserve auction and it can only become higher if more people want to bid on it.
There's a good explanation in the FAQ section of Godaddy:
What is Push to Auction?
Push to Auction is a feature at GoDaddy Auctionsยฎ that lets you convert a domain name listing from Offer/Counter-Offer to a 7-Day Public Auction.
When you receive an offer for a domain name, you can either accept it, decline it, provide a counter offer, or push the listing to auction. If you decide to push your Offer/Counter Offer listing to auction, the buyer's offer becomes the opening bid in the 7-Day Public Auction. Because we convert the offer to a bid, you cannot cancel the listing or add a reserve price after you Push to Auction.
Once you Push to Auction, other GoDaddy Auctions members can bid on the listing. To make tracking easier, the listing ID does not change.
If there are multiple offers for a listing when you Push to Auction, the highest bid becomes the opening bid, and we cancel all pending offers. We add the domain name to the Watching List for each member who made an offer, notified them of the change, and provide a link to the auction.