也许你可以这样做：

def add_contacts(request):
    if request.method == "POST":
        user_id = request.POST['user']   
        client_id = request.POST['client']
        name = request.POST['name']
        email = request.POST['email']
        phone = request.POST['phone']
        mobile = request.POST['mobile']
        role = request.POST['role']
        title = request.POST['title']

        data = Contacts.objects.create(
            user_id=user_id, # call user_id instead "user" only
            client_id=client_id, # call client_id instead "client" only
            name=name,
            email=email,
            phone=phone,
            mobile=mobile,
            role=role,
            title=title,
        )

一旦user和client是Contacts模型上的外键，你就可以像上面一样直接解析id了。
另一个建议是使用get dict函数，以避免一些KeyErrors。我的意思是：

request.POST.get('name', <default_value>) 
# instead
request.POST['name']

这是因为Model需要user和client的示例对象，因为这两个字段都是外键，所以你必须检索它们来创建你的Contacts对象（例如：Client.objects.get(pk=client)）。
从外观上看（分别检索每个字段），您没有使用Django forms。有了这么多的字段和嵌套对象，我会考虑创建一个，让框架来完成它的工作，下面是一个使用ModelForm的例子：
forms.py

class ContactForm(ModelForm):
    class Meta:
        model = Contacts
        fields = [
            "user", "client", "name", 
            "email", "phone", "mobile", 
            "role", "title"
        ]

views.py

def add_contact(request):
    if request.method == "POST":
        form = ContactForm(request.POST)
        if form.is_valid():
            form.save()
            # You probably want to redirect somewhere else
            return redirect("add-contact")
    else:
        form = ContactForm()

    return render(request, "add_contact.html", {"form": form})

add_contact.html

...
<form method="post" action="{% url 'add-contact' %}">
    {%csrf_token%}
    {{form.as_p}}
    <button type="submit" >Create Contact</button>
</form>
...

urls.py

urlpatterns = [
    path('contact/add/', add_contact, name='add-contact')
]