However, the 26-year-old has declined the offer, adding: "It's not like I don't want to play for my country.
"It's always been a great honour for me to represent my country. Just because I'm not playing this tie doesn't mean I will not play Davis Cup."
Bogdanovic must return to the top 100 to receive the highest level of support from the LTA, which has an elite group who are given individual rankings targets based on their experience.
Elena Baltacha, Anne Keothavong, Laura Robson, Mel South, James Ward, Ross Hutchins and doubles duo Colin Fleming and Ken Skupski currently receive top-level support.
Bogdanovic said: "It's frustrating for me and everybody, but it's disrespectful to say [the funding cut] is for [a lack of] effort because I've always gone out and given it my best shot. It's tough out there. If it was easy everybody would be breaking the top 100.
"Talent alone doesn't get you in the top 100. You need the physical and huge mental side of the game. I still feel I need to fulfil my potential and I have someone helping me on the mental side to get where I want to go.
"I found out they obviously didn't believe in me anymore. So I knew I had to go out there and get my ranking up. That's the only thing you can do. That's what I'm trying to do.
"I believe I can be a top 100 player. I still have the passion for the game. I'm 26 but I feel like I've got another six or seven years in my tennis career."
Bogdanovic has been given eight Wimbledon wildcards previously without getting past the first round, earning £75,830.
I'm sure that whatever happens with Alex, whether he qualifies or not he is entirely capable of putting up a great fight
Lawn Tennis Association chief executive Roger Draper
He added: "When you're playing challengers at my level you have to be winning challengers to be earning a little bit of money, otherwise you're hardly covering your costs.
"You have to look at private sponsorship or play club matches, which I did a little bit to give you that extra bit of funding.
"I'm into qualifiers for Wimbledon. If I play well I could qualify. I could be back for Davis Cup in September."
Lawn Tennis Association chief executive Roger Draper says Bogdanovic has the capability to break into the top 100.
"It's absolutely not the end of the road for him," said Draper. "Alex has huge talent and is playing well at the moment.
"The average age of the top 100 players is 27 so in many senses he is in his prime and he does have the talent still to do some real damage.
"He may well break into the top 100 and he is capable of that.
"The committees probably looked at the fact that he has had eight wildcards previously and he went out in the first round every time, although he had very tough draws in all of those matches.
"He will have to go through qualifying now but maybe that's not a bad thing. You get into the tournament confident with a few wins under your belt.
"I'm sure that whatever happens with Alex, whether he qualifies or not he is entirely capable of putting up a great fight."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.