Toronto currently has 30 of its 55 selections under contract. Anthopoulos has until Monday night at midnight ET to sign the remaining players or the Blue Jays will lose their negotiating rights.
"You're always hopeful I guess," Anthopoulos told reporters late Sunday morning. "I'm more realistic about the process -- that you can only do so much.
"You make an offer you feel is fair, if it's good enough, great, [we] would be thrilled to have the players. If it's not good enough, for the top three rounds, we'll get the picks back the following year and we'll move on."
The most prominent unsigned player is top pick Tyler Beede. The 18-year-old right-hander was taken with the 21st overall selection, but the two sides have yet to come to terms.
Complicating matters is Beede signed a letter of intent to play at Vanderbilt University. Beede has said throughout the process that he is very keen on attending school, and at least publicly, he has not changed that stance.
A source indicated to MLB.com's Jonathan Mayo early in the week that Beede and the Blue Jays were not close in contract negotiations. Beede seemed to echo those statements to his local newspaper on Friday.
"We'll see if we come to an agreement, but right now it looks like school is going to be the choice," Beede told the Worcester (Mass.) Telegram and Gazette.
Anthopoulos prides himself on keeping contract negotiations out of the public eye. The second-year GM does not look kindly upon leaks to the press and has been known to sit down with players if they break that code of conduct.
The same rules do not apply to Beede because the young hurler is not under club control. Anthopoulos said it did not bother him in the least that the 6-foot-4, 200-pound hurler talked to the press.
"He's not our player," Anthopoulos said. "We have his rights to negotiate with him -- if he was under contract then I'd have my thoughts on that.
"But I respect the player, I respect the family, they have a decision to make, it's their life, that type of thing. I wouldn't have an opinion on that from a media standpoint, because he's not our player."
Toronto will receive a compensation pick in next year's Draft if it is unable to come terms with players taken in the first three rounds.
The club currently has officially inked outfielder Jacob Anderson (35), right-hander Joseph Musgrove (46) and right-hander Jeremy Gabryszwski (78).
Outfielder Dwight Smith Jr. (53) and right-hander John Stilson (108) also agreed to terms on Sunday afternoon, along with sixth-rounder Anthony Desclafani and eighth-rounder Mark Biggs.
That would leave right-hander Kevin Comer (57), left-hander Daniel Norris (74) and Beede as the four remaining players in the first three rounds that remain unsigned.
The Blue Jays were the most aggressive team in the Draft according to many experts around the game. The club targeted multiple high-ceiling players that were considered tough to sign because of contract demands or commitments to attend a university in the fall.
Despite the high number of guys still without a contract, Anthopoulos doesn't regret taking that approach. If things don't work out, he'll take his chances with next year's crop of players.
"We will never be in a position where we have to sign a player," Anthopoulos said. "The minute you start to think that way, if we said we have to, then we'd be paying these guys a lot more money.
"We drafted them to sign them, but we didn't draft them to sign them at any cost. If 2011 would be our final Draft of all time, then fine, it doesn't matter. But every Draft and every signing impacts the following year's signing. If you are reckless and irresponsible in '11, then I'll hear about it in '12."