Stephen Flynn told the BBC he could not give any commitment as to whether the deadline would be met.
However, the MP said "everything possible" was being done to ensure this was the case.
Mr Flynn said the party was having problems finding new auditors after the previous company resigned in September.
Accountancy firm Johnston Carmichael, which had worked with the party for more than a decade, said the decision was taken after a review of its clients.
However, First Minister Humza Yousaf confirmed he only found out about it when he took on his new role at the end of March.
And Mr Flynn has told BBC Scotland he only learned of the situation in February.
It comes amid the ongoing police investigation into the SNP's finances, which saw its former chief executive Peter Murrell and treasurer Colin Beattie arrested earlier this month.
Both men were released without charge pending further inquiry.
Separate accounts need to be submitted for the Westminster group by 31 May in order to receive "Short Money" - public funding for opposition parties to carry out their parliamentary work. The SNP is in line for about £1.2m.
Mr Flynn told BBC Radio's Good Morning Scotland programme: "I thought it would be a relatively straight forward process to secure new auditors but that's proven not to be the case."
He said this was partly due to the fact that the financial year was nearing its end as well as the overall challenges in the party's finances.
When asked if the party would lose its Short Money if the deadline was not met, Mr Flynn said: "As I understand it, that would be the case, yes."
He described it as a "situation which is in a state of flux" and added: "I wouldn't want to incur any concern amongst staff that we aren't going to be able to meet our deadlines."
Mr Flynn said he only found out by email on 10 February that the party's auditors had resigned in September.
This was despite the SNP's former Westminster leader Ian Blackford last week saying that all relevant information was handed over to Mr Flynn during the changeover in December.
Mr Flynn said "there may well have been discussions between other people" but reiterated that he was only fully informed of the situation on 10 February.
"I became fully aware of the situation in February," he said. "I received an email from a finance officer who advised me that back in September the party's auditors had opted not to continue and we needed to find our own.
"So since then we've been in the process of trying to find our own because it's important that we are able to undertake our commitments in that regard."
First Minister Humza Yousaf has said that appointing new auditors was one of his "major priorities" and has ordered a governance and transparency review.
Police Scotland launched its Operation Branchform investigation in July 2021 after receiving complaints about how more than £600,000 of donations earmarked for independence campaigning were spent.
Questions were raised after accounts showed the SNP had just under £97,000 in the bank at the end of 2019, and total net assets of about £272,000.
Last year it emerged that former SNP chief executive Peter Murrell, who is married to former first minister Nicola Sturgeon, gave a loan of more than £100,000 to the SNP to help it out with a "cash flow" issue after the last election.
He was arrested two weeks ago at the couple's home in Glasgow before also being released without charge pending further inquiries.
He had resigned as SNP chief executive last month after taking responsibility for misleading statements about a fall in party membership.
Treasurer Colin Beattie has now also stepped down. He was also arrested and released without charge as part of the police investigation.