Try adding exceptions in these applications for your Steam folder and restart Steam to see if this resolves your issue. This issue can be exacerbated by anti-virus/internet security applications blocking Steam from doing what it wants to and reading all of Steam's files as the Steam application attempts to verify them.
Anti Virus/Internet Security applications It's worth noting that this is a well discussed issue with no official solutions available, there are plenty of different suggestions on how this can be fixed though and some of these work for some of the people experiencing the problem some of the time, so you'll want to try all of them if you really want to stop Steam from verifying every time you start it.
If the window appears every single time you start up Steam, you're probably experiencing an issue with Steam realising it has updated.
This window is displayed while the update is installed or verification process is running to provide feedback to you (the user) that something is actually happening, rather than staying completely silent, leading you to think that the application hasn't started. Additionally, Steam will occasionally verify all of its files, to ensure that everything is as it should be.
You could try reinstalling Steam on your rig, as mentioned in the comments, maybe something corrupted Steam files? You didn't mention in your question whether you have already tried this.Steam will always check for updates on startup and download and install them automatically to ensure that you're running the latest version. In my opinion you should focus on solving the problem, not working around it. Losing connection has happened to me and letting it reconnect on itself worked for me. Steam actually should really reconnect itself when it's possible, but there is no way to force the process apart from restarting Steam. Sadly, I don't think there is another solution for on-the-go options. Is there anything else that might fix this? Going to community pages, from the main steam client or the overlay, appears to be attempting to connect ("Please wait - Verifying login information…") but then fails ("Error - There was an error communicating with the Steam servers. Steam will automatically retry connection until the service becomes available." which does not seem true.) (It just shows a message: "A connection to the Friends network could not be established.
I've found How can I get the Steam client to reestablish its connection? but the two solutions are restarting, and clicking "log in" from view friends list, which doesn't have a log in button for me. I know I can fix this by just restarting (it's worked before) but this was a time-consuming achievement to get, and if I close the game it's gone, so I'd really like to coax Steam into reconnecting if at all possible.
it sort of thinks it's in offline mode: the overlay says "Friends is not available in offline mode." and trying to install a game results in "Games cannot be installed while Steam is in offline mode.".it sort of thinks it's in online mode: the client menu has a "go offline…" option, which doesn't help – it requires restarting).