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“WordPress cannot establish a secure connection to WordPress.org. Contact the server administrator.”

Please ask your host if there’s a firewall blocking outbound connections.

Perhaps. Or you may have a firewall set for that site that’s not set correctly.

Do you have access to a command line shell via SSH? If so, login to your host and try

curl -v https://api.wordpress.org

When I do this, I see this… What do you see?

sdstern@sds-desk ~]$ curl -v https://api.wordpress.org
*   Trying 198.143.164.251:443...
* Connected to api.wordpress.org (198.143.164.251) port 443 (#0)
* ALPN, offering h2
* ALPN, offering http/1.1
* successfully set certificate verify locations:
*  CAfile: /etc/pki/tls/certs/ca-bundle.crt
*  CApath: none
* TLSv1.3 (OUT), TLS handshake, Client hello (1):
* TLSv1.3 (IN), TLS handshake, Server hello (2):
* TLSv1.3 (IN), TLS handshake, Encrypted Extensions (8):
* TLSv1.3 (IN), TLS handshake, Certificate (11):
* TLSv1.3 (IN), TLS handshake, CERT verify (15):
* TLSv1.3 (IN), TLS handshake, Finished (20):
* TLSv1.3 (OUT), TLS change cipher, Change cipher spec (1):
* TLSv1.3 (OUT), TLS handshake, Finished (20):
* SSL connection using TLSv1.3 / TLS_AES_256_GCM_SHA384
* ALPN, server accepted to use h2
* Server certificate:
*  subject: CN=*.wordpress.org
*  start date: Nov 24 00:00:00 2021 GMT
*  expire date: Dec 25 23:59:59 2022 GMT
*  subjectAltName: host "api.wordpress.org" matched cert's "*.wordpress.org"
*  issuer: C=GB; ST=Greater Manchester; L=Salford; O=Sectigo Limited; CN=Sectigo RSA Domain Validation Secure Server CA
*  SSL certificate verify ok.
* Using HTTP2, server supports multiplexing
* Connection state changed (HTTP/2 confirmed)
* Copying HTTP/2 data in stream buffer to connection buffer after upgrade: len=0
* Using Stream ID: 1 (easy handle 0x5601d4581ce0)
> GET / HTTP/2
> Host: api.wordpress.org
> user-agent: curl/7.79.1
> accept: */*
> 
* TLSv1.3 (IN), TLS handshake, Newsession Ticket (4):
* TLSv1.3 (IN), TLS handshake, Newsession Ticket (4):
* old SSL session ID is stale, removing
* Connection state changed (MAX_CONCURRENT_STREAMS == 128)!
< HTTP/2 302 
< server: nginx
< date: Wed, 27 Apr 2022 14:57:25 GMT
< content-type: text/html; charset=utf-8
< location: https://developer.wordpress.org/rest-api/
< x-frame-options: SAMEORIGIN
< x-nc:  ord 6
< 
* Connection #0 to host api.wordpress.org left intact
[sdstern@sds-desk ~]$