Strikers Andy Carroll and record signing Andre Ayew are both injured, while the Croatian admits he was close to signing AC Milan's Carlos Bacca.
The Colombian opted to stay in Italy, with Bilic unwilling to chase the forward because he says there are other targets "calling me every day".
Bilic added the deal became "a saga".
"I was the one who called it off three weeks ago in Austria," he added. "I wanted him, of course, but I don't want to have to drag players to West Ham.
"We are not Real Madrid or Manchester United but we are a good club and a proud club with players who are proud to play for us. We are not going to beg anyone to play for us."
Bilic says Bacca is "on great money and wants to stay on that level", but hopes West Ham can attract players of that quality in the future, with Bony an option.
The 27-year-old joined Manchester City in January 2015 and has scored 10 goals in 20 starts, but is yet to play this term.
The Ivory Coast international, who signed for Swansea in 2013 from Dutch side Vitesse Arnhem, was not included in manager Pep Guardiola's squad for the Champions League qualifier with Steaua Bucharest.
"He is a good player," Bilic added. "He is an option because he plays in the position where, even before the injuries, we tried to sign a player.
"We have a couple of players we are trying to do a deal with."
Bilic was speaking at a press conference before Thursday's Europa League qualifier second leg against Astra Giurgiu at London Stadium.
The tie stands at 1-1, with West Ham bidding to join Premier League rivals Southampton and Manchester United in Friday's draw for the group stages.
"We are not going to protect the away goal," Bilic said. "It gives you an advantage but you cannot hang on for 90 minutes. We will play a normal game."
Subscribe to the BBC Sport newsletter to get our pick of news, features and video sent to your inbox.
West Ham manager Slaven Bilic says he is interested in signing Manchester City forward Wilfried Bony, but will not "beg" anyone to join the club.