The partnership was first reported by The Wall Street Journal, citing unknown sources.
Goldman has been incrementally pushing into consumer banking products.
It does personal loans and savings accounts under the brand Marcus, and has been looking at introducing other consumer products.
A co-brand credit card is when a company partners with a bank to create a credit card that will spur loyalty and spending with the company.
Details of the partnership are scarce, but Apple has indicated for a while that it intends to grow its payment platform, Apple Pay.
Currently only available on Apple devices, expanding into the physical realm will enable the tech company to achieve their ambition of replacing cash and plastic.
It would also allow Apple to serve more customers in a greater capacity.
A credit card would allow devoted iPhone customers to use Apple pay when purchasing items from a place that does not yet accept contactless or digital payment.
The Wall Street Journal did report that the marriage of the two companies could allow Goldman to offer in-store loans to customers buying Apple products.
There are hundreds of co-brand partnerships, with brands large and small, and it's an intensely competitive business for credit card companies.
Apple is a huge and iconic brand, and it would be a symbolic success for Goldman as it enters the consumer credit card market, industry experts claim.
"It's not going to put them on the map when it comes to market share, but it's definitely a strong customer base you want to have," said Brian Riley with Mercator Advisory Group, which specialises in consulting with companies looking to do co-brand credit cards.
This is the second time Apple and Goldman have been mentioned together as potential partners.
The Journal reported earlier this year that Goldman was looking to take over Apple's iPhone financing program from Citizens Bank.
Apple, which still gets most revenue from the iPhone, reported a hefty rise in earnings during the first three months of this year, alleviating worries about the iPhone's prospects and a hit from US-China trade tensions.
Analysts praised a big jump in revenues in Apple's services business, which is seen as an important element of diversification away from having revenues tied to gadgets.
A 31 per cent rise in services to $9.2 billion ($13.2b) was credited to revenue gains from Apple Pay, Apple Music and other programs.
Goldman Sachs Bank USA last month acquired personal finance startup Clarity Money. Financial terms of the deal were not disclosed.
Free-to-use Clarity Money is a leading app that uses artificial intelligence to help people manage their finances with personalised, "actionable insights" such as saving money or finding better credit cards, according to Goldman Sachs.
"Consumers want a better way to manage their finances," Goldman Sachs' chief executive Stephen Scherr said in a release.
Goldman Sachs and Apple both declined to comment on the report, which cited unnamed people close to the matter.
- with AP
