Spiegel joins on a one-month loan deal.
"I've watched him play a couple of times and we've been given a lot of footage to review as well," Carlisle goalkeeping coach Tony Caig said.
"He's a good size, he's mobile and vocal, and he's a Swiss Under-21 international so he has good pedigree."
He added: "West Ham are looking for him to make that next step and an opportunity to get out and play some football provides that opening for him."
For the big stories and the banter follow us on Twitter: @tribalfootball
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!